Uniqueness conjecture for k-symmetric multiplicatively stable distributions

Let $k>2$. A probability measure is $k$-symmetric if it has the $k$-fold rotational symmetry used in the paper, and $\boxtimes$ denotes free multiplicative convolution. For $0<\alpha\leq 1$, let $\sigma^k_\beta=w_k\boxtimes\nu_\alpha$, where $\beta=\frac{k\alpha}{\alpha+k-k\alpha}$, as in Theorem 1.

Uniqueness conjecture. The measures $\sigma^k_\beta$ are the only $k$-symmetric $\boxtimes$-stable distributions.

The conjecture asserts that the explicitly constructed family exhausts the $k$-symmetric multiplicatively stable laws when $k>2$. Its resolution is not given in the supplied text.