The injective Yamanouchi-word conjecture for dual equivalence components

Let $b4$ be a diagram with $n$ cells. Write $a3am(n)$ for the set of standard Yamanouchi words of length $n$, $a3am_b4(n)$ for the subset associated with $b4$, and let $c1$ denote the signature map on the components of $bca_b4$. For a component $c2=(V,c1,E)$ of $bca_b4$, let $F_{c1(v)}(X)$ be the fundamental quasisymmetric function indexed by the signature of $v$. The injective Yamanouchi-word conjecture. There exists an injective function

$$f_b4\colon \operatorname{SYam}(n)\hookrightarrow S_n$$

fixing $a3am_b4(n)$ and preserving $c1$ such that, for every component $c2=(V,c1,E)$ of $bca_b4$,

$$\sum_{v\in V} F_{c1(v)}(X)=\sum_{\lambda\vdash n}\left|\left\{c0\in\operatorname{SYam}(\lambda):f_b4(c0)\in V\right\}\right|s_\lambda.$$

This would give a Schur-positive expansion for the quasisymmetric functions associated with all components of $bca_b4$, by encoding their vertices through Yamanouchi words and Schur multiplicities.