This is the first I'd heard about this rumor. But after a quick Google search this is what I found specifically:
"Monster Hunter game" can mean a lot of things, but what people want is mainline Monster Hunter.
For the 3DS, Nintendo made a deal with Capcom to prevent mainline Monster Hunter coming to all other platforms (most importantly, PS) for the duration of a generation (2011-2017). That left the door open for Monster Hunter Frontier and other spinoffs if Capcom wanted to.
I consider it very probable that Sony did the same thing, starting as soon as Nintendo's contract with Capcom ended. Except that Sony specifically targeted Nintendo alone, hence why Monster Hunter World could still release on Xbox and PC. That kind of exclusivity deal comes cheaper than outright exclusivity and for Sony it only ever was about Nintendo to begin with.
In any case, there's a chance that a Monster Hunter spinoff could release on Switch, perhaps something like Monster Hunter Stories for the 3DS which was a different kind of RPG set in the Monster Hunter universe. But mainline Monster Hunter shouldn't be expected.