Two-moment criterion for normal convergence in a fixed eigenspace

Let $k,l\geq 2$ be distinct positive integers, and let $\{X_n\}_{n\geq 1}$ be a sequence of eigenfunctions lying in the same eigenspace of a Markov generator $\mathrm{\mathbf L}$ satisfying assumptions (a)--(b)--(c). Let $N\sim\mathcal{N}(0,1)$. Two-moment normality conjecture. As $n\to\infty$, the following are equivalent: $X_n\stackrel{\mathrm{law}}{\longrightarrow}N$; and

$$\mathbb{E}[X_n^{2k}]\to\mathbb{E}[N^{2k}]\quad\text{and}\quad\mathbb{E}[X_n^{2l}]\to\mathbb{E}[N^{2l}].$$

The claim would generalize the Nualart--Peccati criterion by replacing convergence of a single suitable moment with convergence of two even moments. The paper exhibits several pairs $(k,l)$ for which the relevant polynomial decomposition has positive coefficients, but explicitly states that the general assertion could not be proved and may nevertheless be true.