Cain, A. J., and A. Malheiro. "Deciding conjugacy in sylvester monoids and other homogeneous monoids." Int. J. Algebra Comput.. 25 (2015): 899-915. AbstractWebsite

We give a combinatorial characterization of conjugacy in the sylvester monoid, showing that conjugacy is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids.