The deterministic context-free word problem conjecture for special monoids

Let $M=\operatorname{Mon}\langle A\mid w_i=1\ (1\leq i\leq p)\rangle$ be a finitely presented special monoid. A monoid has deterministic context-free word problem when its word problem is a deterministic context-free language. Deterministic context-free word problem conjecture. $M$ has deterministic context-free word problem if and only if it is isomorphic to a free product $G*F$ of a context-free group $G$ and a free monoid $F$. This extends the known structural obstruction provided by the bicyclic monoid. The forward implication is equivalent to the conjecture of Brough, Cain, and Pfeiffer that the bicyclic monoid does not have deterministic context-free word problem; the reverse direction also depends on whether deterministic context-free word problems are closed under free products, which is open.