For an arbitrary square matrix $L$ whose symmetric part $\frac{L+L^\mathrm{T}}{2}\succcurlyeq0$ is positive semidefinite, can one always find a decomposition $L=U^\mathrm{T}QU$ where $U=U^\mathrm{T}\succcurlyeq0$ and asymmetric $Q$ has $\frac{Q+Q^\mathrm{T}}{2}\succ0$?
The answer seems to be positive, but I still would like to ask about it.
Note, here, that '$L$ is asymmetric' means $L\neq L^\mathrm{T}$.