All matrices are real.
Let $P$ be any stochastic matrix (i.e. square, non-negative, rows summing to one).
Let $\Xi$ be a diagonal matrix containing on its diagonal any left eigenvector of $P$ corresponding to eigenvalue one.
Let $\Phi$ be any matrix (we can assume independent columns, but I don't think this is necessary). Typically, $\Phi$ has much less columns than rows.
Let $\| \cdot \|_2$ denote the matrix norm induced by the vector norm $L_2$.
Prove the following statement:
$\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$
In the above $(\cdot)^\dagger$ denotes the Moore-Penrose pseudoinverse. Note that for ergodic $P$s, it becomes equivalent to the ordinary inverse.
In case this is not true, please provide a counterexample.
Even if you could help me just with the ergodic case, I'd be really grateful.
Edit: thanks to @loupblanc, I discovered that my statement was actually incorrect.