The refined Narayana Lie-idempotent conjecture

Let $\mathsf{F}_t$ be the parameterized series, let $\mathsf{F}_{n,t}$ be its homogeneous component of degree $n$, let $\operatorname{\mathbf{ca}}_{n,t}$ be the corresponding $t$-Narayana fraction, and let $\mathfrak{S}_n$ be the symmetric group. Refined Narayana Lie-idempotent conjecture. In the symmetric group ring of $\mathfrak{S}_n$, one has

$$\mathsf{F}_{n,t}\cdot\mathsf{F}_{n,t}=n\operatorname{\mathbf{ca}}_{n,t}\mathsf{F}_{n,t}.$$

This would make the homogeneous components parameterized Lie idempotents up to the scalar $n\operatorname{\mathbf{ca}}_{n,t}$. The identity has been checked through $\mathfrak{S}_6$, but remains conjectural in general.