The eigenvector generation conjecture for the second-kind interval transform

Let $\operatorname{Pyr}:u\mapsto M(1,u)$ be the pyramid operator, let ${\mathcal L}$ be the lift operator sending $u$ to $(a-b)u$, and let $S_{\mathbb Q}\langle a,b\rangle_n$ denote the specified degree-$n$ subspace of homogeneous $ab$-polynomials.

Eigenvector generation conjecture. For each $n\geq 1$, a generating set of $S_{\mathbb Q}\langle a,b\rangle_n$, consisting only of eigenvectors, may be obtained by applying all possible $n$-fold compositions of $\operatorname{Pyr}$ and ${\mathcal L}$ to $1$.

This conjecture proposes a complete eigenvector-based generation procedure for the relevant subspace, paralleling the construction used for the Tchebyshev operator of the second kind. Its status is not resolved in the supplied source.