Crosscut criterion for extending isotopies of plane compacta

Let $X$ be a plane compactum and let $h:X\times[0,1]\to\mathbb{C}$ be an isotopy starting at the identity. A crosscut $Q$ of a complementary domain $U$ of $X$ is an arc in $U$ whose endpoints lie on the boundary of $U$ and whose interior lies in $U$.

Crosscut extension conjecture. The following are equivalent:

1. $h$ extends to an isotopy of the entire plane.
2. For each $\varepsilon>0$ there exists $\delta>0$ such that, for every complementary domain $U$ of $X$ and each crosscut $Q$ of $U$ with $\operatorname{diam}(Q)<\delta$, $h$ extends to an isotopy

$$h_Q:(X\cup Q)\times[0,1]\to\mathbb{C}$$

such that, for all $t\in[0,1]$,

$$\operatorname{diam}(h^t(Q))<\varepsilon.$$

This would remove the uniform-perfectness assumption used in the proved extension theorem and give a necessary and sufficient criterion for extending isotopies of arbitrary plane compacta. The source does not provide evidence that the conjecture has been resolved.