The quadratic-time word problem conjecture for special monoids with virtually free units

Let $M$ be a finitely presented special monoid, generated by $A$, with virtually free group of units. For inputs $u,v\in A^*$, write $n=|u|+|v|$. Quadratic-time word problem conjecture. The word problem for $M$ is decidable in $O(n^2)$ time. Matrix multiplication has the lower bound $O(n^2)$, so this conjecture would make the current reduction-based bound optimal. It is motivated by the conjectured optimal quadratic-time algorithm for matrix multiplication.