The toric Harder–Narasimhan subdivision conjecture

Let $(P,d\sigma)$ be the moment polytope of a polarised toric variety with canonical boundary measure $d\sigma$. A pair $(Q,d\sigma)$ is semistable when $\mathcal{L}_A(f)\geq 0$ for every convex function $f$, where $A$ is the unique affine-linear function satisfying $\mathcal{L}_A(f)=0$ for all affine-linear $f$. Toric Harder–Narasimhan subdivision conjecture. If $(P,d\sigma)$ is not semistable, then it has a subdivision into finitely many polytopes $Q_i$ such that, for the restriction $d\sigma_i$ of $d\sigma$ to the faces of $Q_i$, every $(Q_i,d\sigma_i)$ is semistable. This is the proposed decomposition of an unstable toric variety into semistable pieces; the source presents it as a goal and gives no evidence of resolution.