Honestly, it’s not helpful to make these kinds of speculations unless you have evidence. Really solid evidence. Even if his “name” is on the logs, John Roberts is not an uncommon name (like Alan Dershowitz).
There was a rumor that Roberts was being blackmailed over something having to do with the adoption of their children.
I even find that very unlikely. When he was confirmed, I heard the libs toned down there criticism because their feeling was he could be “turned”. Roberts and his wife like to go DC parties and are therefore susceptible to influence from the Swamp. Justices like Thomas and Alito specifically avoid the DC social scene for just that reason.
The other thing is that Roberts is probably beholden to the Bushes. It’s not exactly a secret that the Bushes are not the conservatives they used to pretend to be.