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The conjugacy problem for positive homogeneously presented monoids

Let $M$ be a positive homogeneously presented monoid ${\langle L \mid R\,\rangle}_{mo}$. If $M$ satisfies the cancellation condition and carries certain particular elements similar to the \emph{fundamental elements} in Artin monoids, then the solvability of the conjugacy problem for $M$ implies that in the corresponding group ${\langle L \mid R\,\rangle}$. In addition to these conditions, if $M$ satisfies the LCM condition (i.e. any two elements $α$ and $β$ in $M$ admit the left (resp.~right) least common multiple), then the solution to the conjugacy problem for $M$ is known. We will give two kinds of examples that do not satisfy only the LCM condition. For these examples, we will give a solution to the conjugacy problem by improving the method given by E. Brieskorn and K. Saito.