Asymptotic regularity conjecture for initial ideals of binomial edge ideals

Let $G$ be an enumerated graph. Let $\ell$ be the length of the longest induced path in $G$, and let $\ell_{<}$ be the length of the longest admissible path in $G$. Write $\operatorname{gin}(J_G)$ for the generic initial ideal of the binomial edge ideal $J_G$, and $\operatorname{in}_{<}(J_G)$ for its initial ideal with respect to the indicated enumeration-dependent order. Asymptotic regularity conjecture.

$$\widehat{\operatorname{reg}}(\operatorname{gin}(J_G))=\ell,$$ $$\widehat{\operatorname{reg}}(\operatorname{in}_{<}(J_G))=\ell_{<}.$$

These equalities are motivated by lower bounds from induced and admissible paths. The authors state that they are not aware of any graph for which equality fails, but the conjecture remains unresolved in the supplied text.