Almost-everywhere zero-measure conjecture for random Julia sets

For each $u$, let $f_u^n$ denote the corresponding iterated random rational map, and let $\mathcal J_u$ be its random Julia set. Lebesgue measure is taken on the relevant complex sphere.

Almost-everywhere zero-measure conjecture. For Lebesgue almost all $u$, $\mathcal J_u$ has zero Lebesgue measure.

The source suggests approaching this through postcritical sets, Koebe estimates, and parameter-elimination techniques, and notes that the analogous statement for the logistic family is known. The conjecture remains unresolved in the stated setting.