Conjecture on block-size ratios for existence of the limiting spectral distribution

Let $\mathbf{A}_n$ be a random matrix with partition $V_1,\ldots,V_m$, where $\nu_i(n)$ denotes the corresponding block proportions, and suppose

$$\lim_{n\rightarrow\infty}\max\{\nu_1(n),\ldots,\nu_m(n)\}>0.$$

Block-size ratio conjecture. If the limiting spectral distribution (LSD) of $\mathbf{A}_n$ exists, then

$$\lim_{n\rightarrow\infty}\frac{\nu_i(n)}{\nu_j(n)}=1$$

for all $i,j\in[m]$.

The conjecture proposes that, under the stated nondegeneracy condition, existence of an LSD forces all block proportions to become asymptotically equal. The surrounding discussion establishes this behavior in special cases, but the general assertion is presented as a conjecture.