Regularity conjecture for strongly elliptic operators

Let $\mathcal L\in\operatorname{\mathcal S\mathcal E}$, and let $G\subset\mathbb C$ be a bounded simply connected domain. The domain $G$ is $\mathcal L$-regular when the associated $\mathcal L$-Dirichlet problem has the regularity property required in the source. Regularity conjecture for strongly elliptic operators. Every bounded simply connected domain $G\subset\mathbb C$ is $\mathcal L$-regular.

This conjecture is presented as a plausible result whose proof would clarify the general case of the inverse approximation statement. The source notes that the corresponding regularity assertion is known for operators with complex-conjugate characteristic roots, but leaves the general strongly elliptic case open.