The maximal stably irreducible family generation conjecture

Fix a Weyl group element $w\in W$, write $m=\ell(w)$ for its length, and let a stably irreducible family be a set of irreducible modules closed under irreducible convolution up to grading shift. Restrict to $w$-adapted representations. Maximal family generation conjecture. Every maximal stably irreducible family of $w$-adapted representations is $m$-generated. The conjecture predicts that maximal stably irreducible families model the clusters of $\mathcal{A}_v[N^w]$, with atomic modules corresponding to cluster variables.