Equality in the dual F-signature formula for Segre products of polynomial rings

Establish that for every integer t ≥ 2 and all positive integers r1 ≤ ··· ≤ rt, the dual F-signature sdual(S_{r1,...,rt}) of the Segre product S_{r1,...,rt} := #_{i=1}^t k[x_{i,1},...,x_{i,r_i+1}] equals the sum over z = (z1,...,z_{t−1}) in D of the product over i = 1,...,t−1 of (rt + z_i)/r_i times sgen(M_z), where D = { (y1,...,y_{t−1}) ∈ Z^{t−1} : −rt ≤ y_i ≤ 0 for all i ∈ {1,...,t−1}, and −r_j y_i − y_j ≤ r_j for all i, j ∈ {1,...,t−1} }, and sgen(M_z) denotes the generalized F-signature of the conic divisorial ideal M_z in the toric ring that realizes S_{r1,...,rt}.

Background

The paper derives an explicit upper bound for the dual F-signature of Segre products of polynomial rings (Theorem 4.3), expressed as a sum over conic divisorial ideals M_z indexed by lattice points z in a set D, weighted by products of (rt + z_i)/r_i and the generalized F-signatures sgen(M_z).

The authors prove that this upper bound is attained (i.e., equality holds) in several families, including when all but the first r_i are equal (Proposition 4.5) and when all r_i are equal (Proposition 4.4). Motivated by these results, they conjecture that the equality holds in full generality, thereby characterizing sdual(S_{r1,...,rt}) exactly by the proposed sum.