The degree-2 real-root counting conjecture for rigid rank 2 modules

Assume $3\le k\le n/2$. Let $J_{k,n}$ be the associated Kac–Moody algebra, let $\alpha$ be a real root of degree $2$, and let $\varphi$ be the map from indecomposable modules to roots. Degree-2 counting conjecture. For every such real root $\alpha$, there are exactly two non-isomorphic rigid indecomposable rank $2$ modules $M_1$ and $M_2$ satisfying

$$\varphi(M_1)=\varphi(M_2)=\alpha.$$

Consequently, the total number of rigid indecomposable rank $2$ modules corresponding to real roots is

$$2{n \choose 6}{n-6\choose k-3}.$$

The formula comes from choosing six labels for the $3$-interlacing data, $k-3$ common labels, and an ordering of the two subsets; it depends on the preceding tightly 3-interlacing conjecture and is presented as an upper-bound count that would become exact if that conjecture holds.