Parfenov–Fedotov approximation conjecture for strongly elliptic operators

Let $\mathcal L\in\operatorname{\mathcal S\mathcal E}$, and let $X$ be a compact set in $\mathbb C$. The spaces $P_{\mathcal L}(X)$ and $C_{\mathcal L}(X)$ denote, respectively, the closures on $X$ of the relevant classes of solutions and continuous functions associated with $\mathcal L$. A compact set whose complement is connected and locally connected is a Carathéodory compact set. Parfenov–Fedotov approximation conjecture.

$$P_{\mathcal L}(X)=C_{\mathcal L}(X)\quad\text{if and only if}\quad X\text{ is a Carathéodory compact set}.$$

The forward implication is the interesting open direction, while the converse is the known sufficient approximability result for strongly elliptic second-order operators. The claim was posed in Parfenov and Fedotov, and it fails for general non-strongly elliptic operators.