Is $S^TD^2S\succcurlyeq(S^TDS)^2$ true if $S^TS=I$ and $D\succcurlyeq0$ diagonal?

Question

$S^TS=I$ and $D\succcurlyeq0$ is a diagonal matrix. Is it true that $S^TD^2S\succcurlyeq(S^TDS)^2$ ?

Simple case: If $S=s$ is a unit norm vector, i.e. $s^Ts=1$, then by Cauchy-Schwartz we have :

$(s^TDs)^2\leq(s^TD^2s)(s^Ts)=s^TD^2s$ .

Ps: This is a research question I encountered right now, not a homework/exam problem. Any hint/suggestion/solution is highly appreciated.

Answer 1

Yes. Since $S^TS=I$, $SS^T$ is an orthogonal projection and so is $I-SS^T$. Therefore $I-SS^T$ and in turn $S^TD^2S-S^TDSS^TDS=S^TD(I-SS^T)DS$ are positive semidefinite.