Vasyunin's conjecture on high-dimensional cyclic quotient families

Let $d\geq 5$, and let a one-parameter family of Gorenstein cyclic quotient singularities have dimension $2d+1$ and Shokurov minimal log-discrepancy $d$. The corresponding points are considered in the torus $T^{(2d+1)}$, whose coordinates may be permuted.

Vasyunin's conjecture. Up to a permutation of the coordinates in $T^{(2d+1)}$, the corresponding points lie in the subtorus

$$x_1+x_2=1.$$

Equivalently, there are no functions $F(S,c_n;x)$ taking only the values $0$ and $1$ for $d\geq 5$.

This conjecture expresses the observed absence of further $0$–$1$ step-functions in Vasyunin's computations. The source does not provide a proof or a resolution.