The toric fixed-locus reducedness conjecture for PTq moduli on Calabi–Yau 4-folds

Let $X$ be a toric Calabi–Yau 4-fold and let $\curly P=\curly P_v^{(q)}(X)$ for $q\in\{-1,0,1\}$. Toric fixed-locus conjecture. If $\curly P^{T_X}$ is zero-dimensional, then $\curly P^{T_X}=\curly P^T$ and it is reduced. The conjecture predicts that, whenever the Calabi–Yau torus fixed locus is zero-dimensional, it captures the full torus fixed locus and has no nonreduced structure. The paper notes that evidence is provided by cases proved in a stated theorem, but leaves the general claim unresolved.