This is a tough one to tease out even with access to the Soviet era records.
The major proponent of the idea, historian Paul Kengor, is supposed to be a serious scholar, though until diving into his chapter on John Hinkley's March 30, 1981 assassination attempt and the background of what was happening in Poland in March 1981, I had not heard of him.
Here is some of Professor Kengor's May 2017 book "A Pope and a President...."
On Sunday, March 29, 1981, the situation at Field Station Berlin was particularly tense. The station’s intelligence technicians heard an alarming communication between two high-ranking Soviet military officers. The technicians already knew that the Soviets had as many as eighteen divisions arrayed along the Polish border—more troop strength than the entire U.S. Army boasts today. But they did not know the force status of those troops. Now, based on this intercepted communication, the status was clear: the Soviet divisions were going to the “highest state of combat readiness.” Moscow appeared ready to invade Poland. The long-feared crackdown on the Solidarity movement and Pope John Paul II’s countrymen seemed at hand. Field Station Berlin monitored the Soviet chatter closely that day and into the next. The Soviet channels were buzzing with information. And then, almost in an instant ... nothing. The chatter stopped.....
That month the Soviets, along with their Warsaw Pact allies were conducting a large scale war game, SOYUZ 81, scheduled to run for a month: March 17-April 7, inside Poland.
At the exact same time Poland's Solidarity movement was rising and getting more militant, calling for a general strike in response to violence against some of its protesting members. The first, "Warning" strike was held on March 27, with at least 12 million workers laying down their tools for four hours.
The Polish communist government and the Soviets were very concerned, this was the largest strike ever seen in a Warsaw Pact country and Solidarity was promising an even bigger action for March 30, an all-out general strike to shut-down the entire country.
However as the first link, to Wikipedia, says "On 30 March 1981, the government of Poland reached an agreement with Solidarity."
This is where things get murky. Kengor's story is:
“I AM IN CONTROL HERE”
Here is where the situation gets even more fascinating. According to Jack, the Soviets were still readying to strike against Solidarity on March 30. The incoming chatter had been “buzzing” all through that day, he says, and then that night it “collapsed,” it “practically went dead.” He cannot say when, exactly, Moscow made the no-go decision, but he knows the chatter virtually went silent that night.
According to Jack, what called off the Soviet dogs on March 30 was not prudence, charity, or some other Christian virtue permeating the Kremlin. What called them off was the shooting of Ronald Reagan. He observed that the pronounced “drawdown” in Soviet communications on Poland began just after the news of the assassination attempt broke. Why? Jack suggests that the shooting set in motion a chain of events that gave the Soviets second thoughts about an invasion. Immediately after the assassination attempt, when the shooter’s motives were unknown and no one knew whether he was part of a larger plot, top U.S. officials placed the Strategic Air Command on alert and raised defense readiness. According to Jack, seeing the U.S. military go on alert “unhinged” the Soviets and changed Moscow’s calculus.
But what really convinced the Soviets to call off the invasion, Jack says, was a statement that the Soviets took to be a display of military strength and resolve —but that, ironically, is ridiculed in the United States to this day. The statement was the famous one by Secretary of State Alexander Haig, who in the aftermath of the shooting told the White House press, “I am in control here.”
Haig was well known to the Soviets. A retired four-star general in the U.S. Army, Haig had served as supreme allied commander of NATO forces in Europe as recently as two years earlier. To Moscow, Haig’s televised 3 proclamation was a show of American strength.
To repeat: the shooting of Reagan, by total happenstance, halted Moscow’s devious plans, according to Jack....
So the primary question is: Were the Sov's preparing a full-scale invasion to put down Solidarity? An action that would be greatly facilitated by the ongoing war games?
Kengor goes on to lay out the argument that the wheels were indeed in motion toward this end.
On the other hand studies of the period say that maybe the Soviets decided against the invasion prior to the shooting of the American President and the events of March 30.
If Polish forces are unable to cope with . . . Solidarity, [Wojciech] Jaruzelski hopes to receive assistance from other countries, up to and including the introduction of armed forces.—Report to Moscow from a Soviet emissary in Poland, quoted in Mark Kramer, “Soviet Deliberations During the Polish Crisis, 1980–1981”
The Wilson Center has Kramer's "Soviet Deliberations..." (181 page PDF)We do not intend to introduce troops into Poland. . . . Even if Poland falls under
the control of Solidarity, so be it. . . . We must be concerned above all with our own country.—KGB Chairman Yuri Andropov, quoted in Mark Kramer, “Soviet Deliberations During the Polish Crisis, 1980–1981”
Additionally, if interested a couple other books on the background of the events:
On Soyuz 81: "A Secret Life: The Polish Officer, His Covert Mission, And The Price He Paid To Save His Country"
On A Setup For A Military Crackdown: "U.S. intelligence and the confrontation in Poland, 1980-81"
And back to the Wilson Center for the second half of the headline "The Soviet Non-Invasion of Poland in 1980/81 and the End of the Cold War" (36 page PDF)
The first half of the headline comes from a 2001 CBS News special report.
And that's all I know. Sometimes they don't invade.
Postscript:
Oh, one more thing.
Six years after the assassination attempt President Reagan was giving a speech as part of Berlin's 750th anniversary:
