Shapiro–Shapiro's multiplicity monotonicity conjecture for words

Let $S_{n+1}$ be the symmetric group, let ${\mathbf W}_n$ be the set of words in the relevant alphabet, and let $\operatorname{mult}:S_{n+1}\to\mathbb{N}^n$ be given by

$$\operatorname{mult}_j(\sigma)=(1^\sigma+\cdots+j^\sigma)-(1+\cdots+j),\qquad j\in[\![n]\!].$$

Extend this additively to words by $\operatorname{mult}(w)=\sum_i\operatorname{mult}(\sigma_i)$ for $w=(\sigma_1,\ldots,\sigma_\ell)$, and order $\mathbb{N}^n$ coordinatewise. Shapiro–Shapiro's conjecture. Given $w_0,w_1\in{\mathbf W}_n$, if $w_0\preceq w_1$, then $\operatorname{mult}(w_0)\leq\operatorname{mult}(w_1)$. This gives a numerical necessary condition for the partial order governing the stratification by itineraries; the source describes the question as open and identifies it as essentially equivalent to a conjecture of Shapiro and Shapiro.