The right-ideal characterization of ultrametric-preserving monoids

Let $\mathbf{A}$ be a subset of the monoid $\mathbf{P_{PU}}=(\mathbf{P_{PU}},\circ,1_{\mathbf{P_{PU}}})$, and define

$$\mathbf{X}:=\{(\mathbb{R}^+,g\circ d^+):g\in\mathbf{A}\}.$$

Let $R_{\mathbf{A}}$ be the set of all right ideals of $[\mathbf{A}]_{\mathbf{P_{PU}}}$, and write

$$\overline{R}:=\bigcup_{R\in R_{\mathbf{A}}}R.$$

For a subsemigroup $S$ of a monoid $M$, write $S^{1_M}$ for $S$ together with the identity of $M$ when that identity is not already in $S$.

Right-ideal characterization conjecture. If $1_{\mathbf{P_{PU}}}$ belongs to $\mathbf{A}$, then

$$\overline{R}^{1_{\mathbf{P_{PU}}}}=\mathbf{P}_{\mathbf{X}}.$$

This is posed as a request to prove or disprove a characterization of the monoid generated by the associated ultrametric-preserving functions in terms of the union of right ideals. The supplied text gives no resolution, so the claim remains open.