v-number bounded by regularity for binomial edge ideals

Establish whether, for every finite simple graph G and its binomial edge ideal J_G in the polynomial ring R=K[x_1,...,x_n,y_1,...,y_n], the inequality v(J_G) ≤ reg(R/J_G) holds, where v(J_G) is the v-number of J_G and reg(R/J_G) is the Castelnuovo–Mumford regularity of the quotient ring.

Background

The paper studies the v-number for binomial edge ideals and relates it to other invariants, notably regularity. Prior work by Ambhore et al. formulated a general inequality relating these two invariants and verified it for several specific graph classes (chordal and whiskered graphs), leaving the general case unresolved.

This problem asks for a definitive proof (or counterexample) of the inequality across all finite simple graphs, connecting combinatorial properties of graphs to algebraic invariants of their binomial edge ideals.