Triangularity conjecture for the sharp-transformed S basis of FQSym

Let $\phi$ be the map from permutations to their lexicographic encoding, and let $X_n$ and $Y_n$ be the subsets of permutations appearing in the diagonal classification. Define an order $<'$ on permutations by

$$\sigma <' \tau \Longleftrightarrow \phi(\sigma) <_{\rm lex} \phi(\tau).$$

Triangularity conjecture. The matrix of ${{\bf S}^\sigma}^\sharp$ in the ${\bf S}$ basis is triangular. Its diagonal values are $1$ for elements of $X_n$ and $0$ for elements of $Y_n$.

This concerns the matrix structure of the $\sharp$-transform in the free quasi-symmetric-function algebra. The supplied text gives no evidence that the assertion has been proved or disproved.