Partial-transposition conjecture for pillar entries of Schubert tangent cones

Let $S_n$ be the symmetric group, let $w,w'in S_n$, and let $r(w)$ and $r(w')$ be their rank matrices. Write $\mathcal{T}_w$ for the tangent cone of the Schubert variety associated with $w$, and call an entry of a rank matrix a pillar entry in the sense of the paper. A partial transposition matches pillar entries while either preserving their positions or transposing their positions.

Partial-transposition conjecture. If

$$\mathcal{T}_w=\mathcal{T}_{w'},$$

then the rank matrices $r(w)$ and $r(w')$ have the same number of pillar entries, and for every pillar entry $r_{ij}$ of $r(w)$, either $r'_{ij}$ is a pillar entry with $r'_{ij}=r_{ij}$, or $r'_{ji}$ is a pillar entry with $r'_{ji}=r_{ij}$.

The conjecture proposes that equal tangent cones are detected by matching pillar entries up to independent transpositions. Its converse is false, while the stated implication was disproved by examples in the source.