The idempotent classification conjecture for integral Dehn quandles

Let $g\geq 1$, let $\mathcal{D}_g$ be the Dehn quandle of a closed orientable surface of genus $g$, and let $\mathbb{Z}[\mathcal{D}_g]$ be its integral quandle algebra. An element is a convex linear combination of simple closed curves if it is a linear combination with nonnegative coefficients whose coefficients sum to $1$. Idempotent classification conjecture. The idempotents of $\mathbb{Z}[\mathcal{D}_g]$ are precisely the convex linear combinations of pairwise disjoint simple closed curves.

The preceding result proves this description for idempotents of length three, while the conjecture asserts it for all idempotents and every genus. The question concerns the structure of idempotents in the integral algebra associated with the Dehn quandle.