Conjecture on maximal third Chern character of rank two semistable sheaves on the degree-five Fano threefold

Let $X$ be the Fano threefold of index $2$ and degree $5$, with ample generator $H$, and let $S$ and $D$ be as in Proposition. Write

$$\operatorname{ch}(E)=2+cH+dH^2+eH^3$$

for a Gieseker semistable sheaf $E\in\operatorname{Coh}(X)$. For an integer $C=3a+\sum_{i=1}^4b_i$, the divisor $D$ on $S$ is chosen from Proposition.

The maximal third Chern character conjecture. If $c=-1$ and $d=\frac12+\frac C5\ll0$, then

$$e\le-\frac13+\frac{\operatorname{ch}_3(i_*\mathcal O_S(D))}{H^3},$$

and a general sheaf $E$ with maximal $e$ fits into an exact sequence

$$0\to\mathcal O_X(-1)^{\oplus2}\to E\to\mathcal O_S(D)\to0.$$

If $c=0$ and $d=-\frac25+\frac C5\ll0$, then

$$e\le\frac1{30}+\frac{\operatorname{ch}_3(i_*\mathcal O_S(D))}{H^3},$$

and a general sheaf $E$ with maximal $e$ fits into an exact sequence

$$0\to\mathcal U\to E\to\mathcal O_S(D)\to0.$$

This conjecture extends the known cases for rank-two semistable sheaves with maximal third Chern character on $\mathbb P^3$ and on a smooth quadric threefold. It predicts both the sharp bound and the structure of a general sheaf attaining equality in the two indicated asymptotic families.