Expected matrix created from two random orthogonal-projection matrices

Question

Consider an arbitrary finite set of orthogonal-projection matrices (symmetric, idempotent, etc.) in $\mathbb{R}^{n\times n}$.
We draw two matrices $Q,P$ uniformly and i.i.d. from this set.

Question: Is the following expected matrix $\mathbb{E}_{P,Q} \left[I-2P+QPQ\right]$ necessarily positive semi-definite?

Clearly, one such matrix is not necessarily PSD. For instance, $P=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},Q=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$ do not yield a PSD matrix together, but I'm asking about the expectation rather than about an arbitrary instantiation.

Answer 1

Found a counter example.

If we take the set to be $\left\{ \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}, \frac{1}{2}\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix} \right\}$, then we have $\mathbb{E}_{P,Q} \left[I-2P+QPQ\right]= \frac{1}{16}\begin{bmatrix}1 & -5 \\ -5 & 11\end{bmatrix}$, whose eigenvalues are $-0.06694174, 0.81694174$.