Suppose I have a random $m \times m$ matrix $R \sim \mu$ that is possibly singular. Is it true that $ E[R] \propto I$ implies that there exists a scalar $r_{\mu, m}$ such that $E[R^+] = r_{\mu, m}I$, where ${}^+$ denotes the Moore-Penrose pseudo-inverse?

If this is false but there are certain conditions that are needed to make this true, then I'd be interested to know those conditions.

For instance it is true when $R = X^\top X$ and $X$ is $n \times m$ normally distributed (at least I have found references for this when $n\not\in [m-3, m+1]$). Not sure if this is the only non-trivial case.