Equivalence of inner cohomology vanishing and Fano projectivization on Fano varieties

Determine whether, for any Fano variety X of dimension at least 4 (beyond projective spaces P^n and smooth quadrics Q^n), the following two conditions on a rank-2 vector bundle E on X are equivalent: (i) H^i(X, E(t)) = 0 for all integers t and for 2 ≤ i ≤ dim X − 2 (i.e., E has no inner cohomology), and (ii) the projectivized bundle P(E) is Fano.

Background

Theorem 1.9 (Ancona-Peternell-Wisniewski, Malaspina) establishes that for X = P^n or X = Q^n (n ≥ 4), three conditions on a rank-2 bundle E are equivalent: (1) E has no inner cohomology (vanishing H^i(E(∗)) for 2 ≤ i ≤ n − 2), (2) the projectivization P(E) is Fano, and (3) E splits as O(a)⊕O(b) or belongs to a small list of special bundles (including spinor bundles on Q^4).

The authors ask whether the equivalence between the first two conditions—vanishing of inner cohomology and P(E) being Fano—extends to other Fano varieties of dimension ≥ 4, noting that their proof of equivalence on P^n and Q^n proceeds indirectly via the third condition.