Net graph asymptotic regularity conjecture

Let $N$ be the net graph. Its binomial edge ideal is denoted by $J_N$, and write $\widehat{\operatorname{reg}}(J_N)$ for its asymptotic regularity. Net graph asymptotic regularity conjecture.

$$\widehat{\operatorname{reg}}(J_N)=3.$$

The paper presents this as an example expected to have asymptotic regularity different from two, in contrast with the preceding results for the considered classes of graphs. Its status is not resolved in the supplied text.