Let $R$ be the root system of a Weyl group $W$. Let $tilde{R}^+$ be the set of $B$-cosmall roots, i.e. the set of positive roots $alpha$ such that $ell(s_alpha)=2operatorname{ht}alpha-1$. Based on the combinatorics developped in Mare’s paper, Section 3, I have the following conjecture.

**Conjecture.** Let $alpha_1,ldots,alpha_r,gamma_1,ldots,gamma_rintilde{R}^+$ be such that $s_{alpha_1}cdots s_{alpha_r}=s_{gamma_1}cdots s_{gamma_r}$, $$ell(s_{alpha_1}cdots s_{alpha_r})=ell(s_{gamma_1}cdots s_{gamma_r})=ell(s_{alpha_1})+cdots+ell(s_{alpha_r})=ell(s_{gamma_1})+cdots+ell(s_{gamma_r}),,$$ and such that $alpha_1^vee+cdots+alpha_r^vee=gamma_1^vee+cdots+gamma_r^vee$. Then, we have ${alpha_1,ldots,alpha_r}={gamma_1,ldots,gamma_r}$.

I proved that this conjecture is true whenever $r=2$, $alpha_1notperpalpha_2$, $gamma_1notperpgamma_2$. While the general conjecture above might be too optimistic, I would like to see a counterexample preferably for $r=2$, but could not find one, and a counterexample for $r>2$ is also welcome. Any help is appreciated!

**Motivation.** Concerning the motivation for this conjecture, I would like to add that an affirmative solution to the above conjecture would have implications on numerical bounds on three point genus zero Gromov-Witten invariants. For example, in type $mathsf{A}_n$, one might conjecture that $$N_{u,v}^{w,d}leqleftlceilfrac{n}{2}rightrceil!$$ for all $u,v,winmathbb{S}_{n+1}$ and all degrees $d$, where $N_{u,v}^{w,d}$ is the three point genus zero Gromov-Witten invariant of degree $d$ passing through three Schubert cycles parametrized by $u,v,w^*$, in other words, the structure constant of (small) quantum cohomology.