The converse conjecture for F-support containment and row-overlap dominance

Let $A$ and $B$ be skew shapes of the same size. For each positive integer $k$, let $\mathrm{rows}_{k}(A)$ and $\mathrm{rows}_{k}(B)$ denote their row-overlap partitions, and let $\mathrm{supp}_{F}(A)$ denote the set of descent compositions indexing the fundamental quasisymmetric functions occurring in the expansion associated with $A$. Write $\preceq$ for dominance order on partitions. The converse conjecture.

$$\mathrm{supp}_{F}(A) \supseteq \mathrm{supp}_{F}(B) \quad\Longleftrightarrow\quad \mathrm{rows}_{k}(A) \preceq \mathrm{rows}_{k}(B) \text{ for all } k.$$

This conjecture gives a characterization of containment of $F$-supports by dominance of row-overlap partitions and, equivalently, identifies the relevant $F$-support and overlaps posets. It has been computationally checked for all skew shapes of size at most $12$, while the general case remains open.