Prime tableaux from coarsest positroidal subdivisions

Let $T \in \operatorname{SSYT}(k,[n])$ be a tableau with no frozen factors. Write $T=S^{\cup r}$ with $r$ maximal, where $S=\underline{T}$ has no further common frozen factor. A positroidal subdivision of the hypersimplex $\Delta(k,n)$ is coarsest if it admits no nontrivial refinement. Prime-tableau conjecture. If $F_{k,n}(v_T)$ defines a coarsest positroidal subdivision of $\Delta(k,n)$, then $\underline{T}$ is prime, meaning that the corresponding dual canonical basis element cannot be written as a product of two dual canonical basis elements. This would establish the converse direction to the known correspondence for $\operatorname{Gr}(2,n)$; for $k\geq 3$, prime tableaux can correspond to subdivisions that are not coarsest, so only this implication is conjectured.