A word-hyperbolic special monoid with undecidable Diophantine problem

The Diophantine problem for a monoid M is the decision problem to decide whether any given system of equations has a solution in M. In this short note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special monoid M with trivial group of units, and such that the Diophantine problem is undecidable in M. This answers two questions asked by Garreta and Gray in 2019, and shows that the decidability of the Diophantine problem in hyperbolic groups, as proved by Dahmani and Guirardel, does not generalise to word-hyperbolic monoids.