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 $\ell(s_\alpha)=2\operatorname{ht}\alpha-1$. Based on the combinatorics developped in [Mare's paper, Section 3][1], I have the following conjecture.

**Conjecture.** Let $\alpha_1,\ldots,\alpha_r,\gamma_1,\ldots,\gamma_r\in\tilde{R}^+$ 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 $r=2$, $\alpha_1\not\perp\alpha_2$, $\gamma_1\not\perp\gamma_2$. 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 $$N_{u,v}^{w,d}\leq\left\lceil\frac{n}{2}\right\rceil!$$ for all $u,v,w\in\mathbb{S}_{n+1}$ 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.

**Edit.** You are also welcome to give a counterexample to the bounds in the motivation.


  [1]: https://arxiv.org/abs/math/0301257v3