Even after the Checkmate buff I still think the MU is even.
Bombs beat everything of Robin's that isn't Thoron and Link's spam makes it easier for him to win the projectile game because he's that much more mobile when throwing out projectiles (Robin has to stand still for Thunder charging, and can only move a little when casting Arcfire)
I would give Robin the advantage in the air for the most part because of the strength/speed/disjointed range of the Levin sword, which makes her aerial set better than Link's IMO, except for the fact that Link's Uair/Dair will outrange, but those two moves are situational and pretty heavy commitments anyways (Uair not being as bad as Dair of course).
Robin has faster options on the ground than Link but Link's projectiles have more of a presence there than in the air (which is very important for him in this MU IMO) and I feel like Link has a slight range advantage on the ground as well in terms of his Normal set.
Link's Smash attacks overall aren't as heavy commitments as Robin's are and kill earlyish. His F-tilt is a heavier commitment but can actually kill unlike Robin's, U-tilt can kill earlier than Robin's. F-Smash Swing 1 is a relatively low commitment option that confirms into Swing 2 which kills really early and can also hit with a tipper for a stock as well. Link's D-Smash is surprisingly safe, and his grab range is enormous now which allows him to get kills off U-Throw a bit easier simply because he has the range. Robin is also just tall enough to be reasonably poked with Zair.
Both can mess with each other's recoveries pretty easily, Robin's Elwind is a free Dair or anything else for Link which often means a stage spike if Robin is recovering from below the ledge.
Similarly Link is susceptible to the Nairplane because his recovery is rather predictable.
This is all just off the top of my head, though.
IMO if Link tries to go all CQC without his projectiles, Robin will win, but if Link starts throwing out Boomerangs and especially Bombs
intelligently then he has the advantage because he can apply better overall pressure which confirms or at least racks damage towards his kill options.
Granted, it's been a while since I've played in general and since I've played a post-buff Robin, but a lot of this still applies I think.
I would need to play
@Raziek again or another Robin of a similar level now to get a better idea, but that's nigh impossible for me right now because of college.