The rank inequalities for symmetric-group actions on sl3 webs

Let $w$ be a reduced $\mathfrak{sl}_3$ web, let $s_i$ be a simple transposition in the symmetric-group action, let $s_i\cdot w$ be expanded in the reduced-web basis, let $r(w)$ denote rank, and let $\preceq_S$ be the shadow-containment partial order. For every reduced summand $\widetilde{w}$ of $s_i\cdot w$,

$$\widetilde{w}\text{ is a reduced summand of }s_i\cdot w.$$

Rank-inequality conjecture. One has

$$r(\widetilde{w})\leq r(w)+1.$$

Furthermore, if $\widetilde{w}\preceq_S w$, then

$$r(\widetilde{w})\leq r(w).$$

These inequalities are proposed as sufficient local estimates for the rank-separation conjecture, replacing corresponding shadow-containment statements. They remain open.