Cohomology vanishing for symmetric squares of SL2-equivariant instantons on P3

Prove that for every integer m ≥ 1, the bundle F_m = S^2 E_m on P^3 satisfies H^i(F_m(−2)) = 0 for all i, where E_m is the unique SL_2(C)-equivariant rank-2 instanton bundle on P^3 (for the action by binary cubics) with charge C(m+1, 2).

Background

The authors construct Ulrich bundles on Veronese threefolds via symmetric squares of rank-2 instanton bundles and show that for the SL2-equivariant instantons E_m, the bundle F_m = S2 E_m satisfies H*(F_m(d−2))=0 with d=2m+1. This, together with deformation arguments, yields Ulrich bundles in rank 3 for the target Veronese varieties.

They note that establishing the additional vanishing H*(F_m(−2))=0 would allow bypassing a step in the argument (reliance on irreducibility of the instanton moduli). While this vanishing holds for several tested values of m, a general proof is currently lacking.

References

We could actually bypass it by showing $H*(_m(-2))=0$, which we checked for several values of $m$ but failed to prove in general.

— Ulrich ranks of Veronese varieties and equivariant instantons  (2405.12574 - Faenzi et al., 2024) in Introduction (paragraph after Theorem 2)