Symbolic F-splitness of binomial edge ideals for weakly closed graphs

Ascertain whether, for every weakly closed graph G (i.e., a graph on vertices labeled [d] such that if {i, k} is an edge and i < j < k, then at least one of {i, j} or {j, k} is an edge), the associated binomial edge ideal J_G ⊂ k[x_1, …, x_d, y_1, …, y_d] is symbolic F-split.

Background

Weakly closed graphs generalize closed graphs and include several families (such as complete multipartite and caterpillar graphs) that the authors show are symbolic F-split. Despite these advances, the property is not established for the entire class of weakly closed graphs.

The open question seeks to extend symbolic F-splitness beyond specific subclasses and determine whether it holds uniformly across all weakly closed graphs, bridging combinatorial graph properties with F-singularities of their binomial edge ideals.