Uniform local-index conjecture for minimal cancelling tuples

Let $csigma_i$ be singularities with local indices $cell_{csigma_i}$, and let $Q_{csigma_i}$ denote their orbifold contribution terms. Suppose

$$\sum_{i=1}^n Q_{\sigma_i}=0.$$

Uniform local-index conjecture. Then $cell_{csigma_i}=cell_{csigma_j}$ for all $i,j$.

This conjecture asks whether a minimal cancelling tuple can contain singularities of different local indices. It is unresolved and is used to sharpen later results about recovering baskets from Hilbert series.