Generalized symmetry algebra conjecture for the fine Kolmogorov backward equation

Let $u$ satisfy the fine Kolmogorov backward equation $L u=0$, where

$$L:=\mathrm D_t+x\mathrm D_y-x^2\mathrm D_x^2.$$

Let $\hat\Lambda$ denote the Lie algebra of reduced linear generalized symmetries, let $u\partial_u$ be the simplest linear generalized symmetry, and let $\mathrm P^y$, $\mathrm D$, and $\mathrm K$ be the recursion operators of the equation. Write $\Upsilon_{\mathfrak f}$ for the associative algebra generated by these operators.

Generalized symmetry algebra conjecture. Up to equivalence of generalized symmetries and after neglecting the Lie symmetries associated with the linear superposition of solutions, all generalized symmetries of the fine Kolmogorov backward equation are reduced linear generalized symmetries. Moreover, their Lie algebra is generated from $u\partial_u$ by the recursion operators $\mathrm P^y$, $\mathrm D$, and $\mathrm K$, namely

$$\hat\Lambda=\{(\mathrm Q u)\partial_u\mid\mathrm Q\in\Upsilon_{\mathfrak f}\}.$$

The conjecture concerns the still-uninvestigated generalized symmetry algebra of this equation; its proposed description is based on preliminary analysis and is presented as a topic for further research.