Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of weights in $\mathrm{Sym}^n(V)$.

I've seen many allusions to the fact that the analogous statement is true for general compact Lie groups $K$ (i.e., one pushes further forward to a positive Weyl chamber and compares with the irrep distribution), e.g., in Allen Knutson's reply to my last question and in the appendix of Guillemin-Prato's 1990 paper, but could not find an explicit statement of this in the literature.

Do you know whether this statement is true at all, and do you maybe even have a reference?