Weak Bruhat monotonicity conjecture for valid hook configuration counts

Let $S_r$ be the permutations of length $r$, and let $\leq_B$ be the weak Bruhat order on $S_r$, generated by adjacent swaps that replace an ascent by the resulting permutation. For a pattern $\rho$, let $\operatorname{Av}_n(\rho)$ be the permutations of length $n$ avoiding $\rho$, and let $\operatorname{VHC}(\operatorname{Av}_n(\rho))$ be their valid hook configurations.

Weak Bruhat monotonicity conjecture. If $\sigma,\tau\in S_r$ satisfy $\sigma\leq_B\tau$, then

$$\left|\operatorname{VHC}(\operatorname{Av}_n(\sigma))\right|\leq\left|\operatorname{VHC}(\operatorname{Av}_n(\tau))\right|$$

for all $n\geq 1$.

The source says this is known for $n\leq r$ and is motivated by the apparently faster growth of the 321-avoiding case; it does not report a general proof or disproof.