Waldschmidt constants of matroid configurations with non-uniform specialization degrees

Compute explicitly the Waldschmidt constant ẑα(φ_*(I_{Δ(M)})) for matroid configurations when the specializing forms f_1,…,f_n have arbitrary (not all equal) degrees, i.e., develop effective methods to evaluate min_{1≤r≤|E|−rk(M)} (Σ_{i∈U} deg f_i)/r over circuits U of the elongations M^{(r)}.

Background

When all f_i have the same degree δ, the paper proves ẑα(φ*(IΔ))=δ·ẑα(I_Δ). For non-uniform degrees, Corollary 7.3 expresses ẑα(φ*(IΔ)) as a minimum of weighted circuit sums across elongations, but no general computational method is provided.

An explicit, generally applicable computation or simplification in the non-uniform degree case remains open.