Realization conjecture for metric-preserving function monoids

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a metric space $\mathbf{X}$, and let $\mathbf{Am}$ be the monoid of amenable functions. Realization conjecture. The equality

$$\mathbf{P}_{\mathbf{X}}=\mathbf{A}$$

has a solution $\mathbf{X}\subseteq\mathbf{M}$ for every submonoid $\mathbf{A}$ of the monoid $\mathbf{Am}$. The preceding example shows that $\mathbf{Am}$ cannot be replaced by $\mathbf{F}$ in this equality, so the conjecture concerns realization specifically within the amenable-function monoid.