Positive-measure disconnectedness conjecture for random Julia sets

Let $\mathbb T$ be the parameter circle, and for each $u\in\mathbb T$ let $\mathcal J_u$ be the corresponding random Julia set. A subset of $\mathbb T$ has positive Lebesgue measure when its Lebesgue measure is positive.

Positive-measure disconnectedness conjecture. There exists a set of positive Lebesgue measure $\Lambda\subset\mathbb T$ such that $\mathcal J_u$ is disconnected for all $u\in\Lambda$.

This conjecture describes parameter values with disconnected random Julia sets and is stated as a reasonable expectation. The source gives no resolution.