Bipartite graphs with sequentially Cohen–Macaulay binomial edge ideals

Determine all bipartite graphs G for which the binomial edge ideal J_G in S = K[x_1,...,x_n,y_1,...,y_n] is sequentially Cohen–Macaulay.

Background

After establishing that cycles, wheels, block graphs, and certain cones yield sequentially Cohen–Macaulay binomial edge ideals, the authors point out that, unlike the Cohen–Macaulay case where bipartite graphs are completely classified, the sequentially Cohen–Macaulay classification for bipartite graphs remains unsettled.

This question seeks a structural characterization of bipartite graphs whose binomial edge ideals are sequentially Cohen–Macaulay, paralleling known classifications in the Cohen–Macaulay setting.