Dominance-order conjecture for the Fibonacci-word multiplicity matrix

For $n\ge1$, let $\mathbb{YF}_n$ be the set of Fibonacci words of weight $n$. For $u,v\in\mathbb{YF}_n$, let $N_{u,v}$ be the corresponding multiplicity, and arrange these entries in lexicographic order into the matrix $\mathrm{\bf N}_n$. Write $u\succeq v$ for the dominance order on Fibonacci words, and let $\mathcal{h}(w)$ denote the total hike, the number of $2$'s appearing in $w$.

Multiplicity-matrix conjecture. The matrix $\mathrm{\bf N}_n$ is upper triangular. Furthermore, if $N_{u,v}$ is nonzero, then $u\succeq v$ and $\mathcal{h}(u)=\mathcal{h}(v)$. This conjecture predicts a triangularity and dominance structure for the multiplicities arising in the Fibonacci-word expansion of the moments. The source gives no resolution status, so it remains open.