Uniqueness conjecture for infinitely divisible solutions of the size-bias equation

Let $c>1$, and let $U_c$ be the Choquet simplex of probability distributions solving the size-bias scaling relation associated with $c$. Write $c=\exp(\sigma^2/2)$, and let $Z$ be standard normal. A distribution is infinitely divisible if it is infinitely divisible under convolution. Uniqueness conjecture. The only infinitely divisible distribution in $U_c$ is the lognormal distribution of $\exp(\sigma Z)$. The claim singles out the lognormal among the solutions of the size-bias equation; the source gives no resolution.