Characteristic-two Newton-slope criterion for hyperelliptic Jacobians

Let $A$ be a $3$-dimensional abelian variety over a finite field of characteristic $2$, with Weil polynomial having Newton polygon whose first slope is $\frac{1}{3}$; equivalently, in the notation of the source, it satisfies $v_p(u)\geq r$. Characteristic-two slope conjecture. Every such abelian variety contains the Jacobian of a genus-$3$ hyperelliptic curve. In characteristic $2$, the paper proves that $2$-rank-$0$ hyperelliptic Jacobians have first Newton slope $\frac{1}{3}$, while the converse is presented as suggested by inspection of the data and remains unproved.