The absolute-continuity conjecture for multiplicative convolutions of self-similar measures

Let $\mu_1,\dots,\mu_k$ be self-similar measures supported in $(0,\infty)$ with

$$\sum_{i=1}^k\kappa_2(\mu_i)>1.$$

Their multiplicative convolution $\mu_1\cdot \mu_2 \dotsb \mu_k$ is the pushforward of the product measure under multiplication.

Multiplicative-convolution conjecture. The measure $\mu_1\cdot \mu_2 \dotsb \mu_k$ is absolutely continuous with respect to Lebesgue measure.

The conjecture is motivated by the corresponding positive-measure problem for arithmetic products and by the paper's Fourier-decay approach. The supplied evidence resolves it for certain specific self-similar measures, including the middle-third Cantor measure, but not in the stated generality.