Norm of a product of projections in a Banach space

Question

Let $X$ be a Banach space and let $P_1,P_2$ be two projections in $B(X)$, i.e., $P_1^2 = P_1, P_2^2=P_2$.

My question: under what conditions do we have that $\Vert P_1 P_2 \Vert = \sqrt{\Vert P_2 P_1 P_2 \Vert}$? ($\Vert . \Vert$ is the usual operator norm)

Easy cases in which this is true:

1. $P_1 P_2 = 0$.
2. $X=H$ is a Hilbert space and $P_1,P_2$ are orthogonal projections

Is something more general known?