Generic non-jumping-line conjecture for instantons on the Fano threefold $Y_5$

Let $Y$ be the Fano threefold $Y_5$, and let $E$ be an instanton on $Y$. A line is jumping for $E$ if its restriction to $E$ is nontrivial.

Generic non-jumping-line conjecture. The generic line is not jumping.

This conjecture is presented as a possible step toward translating statements about jumping lines into statements about jumping conics. The supplied text gives no resolution, so its status remains open.