Block decomposition conjecture for permutation matrices

Let $S_n$ be the symmetric group, let $U(n)$ be the unitary group, and let $\Pi_n$ be the $n!$-simplex with vertices indexed by $S_n$. For partitions

$$\{1,\dots,n\}=A_1\dot\cup\dots\dot\cup A_m=B_1\dot\cup\dots\dot\cup B_m,$$

with $|A_s|=|B_s|$ for $1\leq s\leq m$, define

$$\mathcal{P}(A,B)=\{\alpha\in S_n:\alpha(A_s)=B_s,\ 1\leq s\leq m\}.$$

If $x=\sum_{\alpha\in\mathcal{P}(A,B)}x_\alpha\alpha$, the block decomposition condition requires the non-zero matrix coefficients of $u_{ij}(x)$ to be supported on $\bigcup_{s=1}^m B_s\times A_s$.

Block decomposition conjecture. There should be a continuous function $u$ from $\Pi_n$ into $U(n)$ such that each vertex is mapped to the corresponding permutation matrix and the block decomposition condition holds for every such pair of partitions.

The conjecture arises in the analysis of the multivariable classification problem for tensor algebras and reduces the conjectured converse to a topological question about $U(n)$. The supplied text does not state whether it has been resolved.