I don’t know, there’s a few holes in that claim anyways. For starters, an internal memo from the CIA indicated that LBJ wasn’t even aware that LHO tried to kill JFK until roughly three days afterward. And then there’s the fact that he tried to get a passport for the USSR and Cuba a couple weeks earlier, also mentioned in the same memo.
I’m pretty sure that the Soviets were indeed involved. One of the FBI memos even indicated that a certain New York charter of the Communist Party was going overtime trying to denounce the whole thing as being a right-wing conspiracy and deny any involvement in the assassination or even that LHO was one of their own.
As the Cold War intensified, Morris and Jack Childs were recruited by the FBI. Under the name Operation Solo, the FBI developed them into top level double agents who operated at the highest levels of the Communist Party in the US and internationally. The Soviets trusted and relied on Morris Childs so much that he was even allotted an apartment in Moscow.
Shrewdly, Hoover kept Operation Solo secret, even from the CIA and every President before Gerald Ford. As it happened, Morris Childs was in Moscow meeting with a key Soviet official when the news arrived of Kennedy's assassination. Covertly, Childs had learned to speak Russian and was privy to Soviet alarm at the possibility that Oswald had been one of their agents. A flurry of urgent consultations proved otherwise. Morris Childs was then told of that and to return to the US to make clear to the US Communist Party that Oswald was not a Soviet agent and to refute any such claims.
Of course, the possibility remains that Castro or a rogue element of the KGB or GRU might have encouraged or directed Oswald as an assassin. Alternatively, as part of a domestic conspiracy, rogue US intelligence officers might have created a false trail implicating the Russians or Cubans and thereby provided a compelling reason for the Warren Commission to avoid pursuing evidence of a conspiracy since doing so could have provoked a nuclear war.
The CIA station chief in Mexico City seems to have suspected something like that, and it has been reported that Kennedy loyalists in the CIA were told that Bobby Kennedy did not want any links to Cuban or Russia pursued. That fits with Bobby's supposed belief that LBJ was at the heart of the conspiracy that had his brother killed.