Isomorphic realization conjecture for submonoids of metric-preserving functions

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{F}$ be the monoid of functions under composition, and let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a metric space $\mathbf{X}$. Isomorphic realization conjecture. For every submonoid $\mathbf{A}$ of the monoid $\mathbf{F}$ there exists $\mathbf{X}\subseteq\mathbf{M}$ such that $\mathbf{P}_{\mathbf{X}}$ and $\mathbf{A}$ are isomorphic submonoids. This is proposed as a weaker replacement for the preceding equality-realization conjecture, since equality realization fails when $\mathbf{Am}$ is replaced by $\mathbf{F}$.