Hypoellipticity characterization for symmetric degenerate elliptic operators

Let $L$ be the operator from the paper's general form, assumed to be symmetric, and let $g(x)\geq 0$. Let $L_1$ and $L_2$ be the one-dimensional operators appearing in $L$, and let the superlogarithmic estimate be the estimate denoted by (superlog) in the source.

Hypoellipticity characterization. The operator $L$ is locally $C^\infty$-hypoelliptic if and only if both $L_1$ and $L_2$ satisfy the superlogarithmic estimate.

This conjecture proposes a necessary and sufficient condition extending the paper's theorem from the specific class of operators treated there to more general symmetric operators $L_1(x)$. The supplied passage does not establish the conjecture or indicate a resolution.