Positive Ricci curvature on fiber bundles

Question

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:

Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold $M$ with fiber $F$, compact structure group $G$ and base $B$. Suppose that: i) $B$ has a metric of positive Ricci curvature; ii) $F$ has a $G$-invariant metric of positive Ricci curvature. Then $M$ carries a metric of positive Ricci curvature.

Both the referee and us are not sure if it is in literature or not. It happens that the conjecture is easily proved using classical arguments. Therefore, we would like to ask if someone knows a reference, or if the conjecture is well known to be true among specialists.

Partial results we could find are:

1) (Nash) https://projecteuclid.org/euclid.jdg/1214434973

2) W. A. Poor, Some exotic spheres with positive Ricci curvature, Math. Ann. 216 (1975) 245-252.

Answer 1

This conjecture is already proved in

Gromoll, Detlef; Walschap, Gerard, Metric foliations and curvature, Progress in Mathematics 268. Basel: Birkhäuser (ISBN 978-3-7643-8714-3/hbk). viii, 174 p. (2009). ZBL1163.53001.

(page 100, Theorem 2.7.3).