Determinant formula for the weak-order raising operator

Let $W_n$ be the weak order on the symmetric group $S_n$, ranked by inversion number, and let $U$ be the order-raising operator defined by

$$U(u)=\sum_{i\,:\,\ell(us_i)=1+\ell(u)}i\cdot us_i.$$

For $0\leq k<\frac12\binom n2$, let $\tilde{D}(n,k)$ be the matrix of the map $U^{\binom n2-2k}\colon \mathbb{Q}(W_n)_k\to\mathbb{Q}(W_n)_{\binom n2-k}$ after dividing every entry by $(\binom n2-2k)!$. The determinant conjecture.

$$\det \tilde{D}(n,k)=\pm\prod_{i=0}^{k-1}\left(\frac{\binom n2-(k+i)}{k-i}\right)^{\#(W_n)_i}.$$

This formula has been verified for all pairs $(n,k)$ with $n\leq 12$ and $k\leq 5$, as well as in a few additional cases. If true, it would establish the required nonsingularity of the raising maps and hence strong Sperner properties for the weak order.