Estimate for positively curved manifolds and simply-connected fibrations

Let $M$ be a closed simply-connected positively curved manifold, and let $F\hookrightarrow M\to B$ be a fibration of simply-connected spaces. Assume that the estimate referred to in the source as Estimate (2) is defined for this fibration.

Conjectured estimate. Estimate (2) is well-defined and holds.

This statement would extend the fibration estimate studied in the paper to closed simply-connected positively curved manifolds. Its resolution is not established in the supplied text.