Compatibility conjecture for valuation ideals under tree morphisms

Let $\Delta_{\ell-1}$ and $\Delta_\ell$ be the value groups appearing in the tight extensions, let $\hat{\mathcal P}”_{\beta\ell}$ and $\hat{\mathcal P}'_{\beta\ell}$ be the corresponding valuation ideals, and let $\phi_j$ be the elements of the cited lemma. For a positive element $\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and a tree morphism $R'\rightarrow R”$ in $\mathcal T$, the valuation ideals should satisfy Compatibility conjecture. The elements $\phi_j$ can be chosen so that

$$\hat{\mathcal P}”_{\beta\ell}\cap\hat R'_{\tilde H'_{2\ell}}=\hat{\mathcal P}'_{\beta\ell}.$$

This compatibility is intended to ensure that the valuation ideals defined on the completed local rings are compatible throughout the directed system of tree morphisms. The source does not provide evidence resolving whether such a choice of the elements $\phi_j$ always exists.