The diagonal quasi-symmetric Hilbert matrix recurrence conjecture

Let ${\rm DQ}_n={\mathbb Q}[{\bf x},{\bf y}]/\mathcal J_n$ be the quotient by the ideal generated by diagonally quasi-symmetric polynomials without constant term, and let $M_n$ be its Hilbert matrix. Thus $M_n$ is an $n\times n$ lower triangular matrix. Hilbert matrix recurrence conjecture. Every diagonal entry is the $(n-1)$st Catalan number,

$$M_n(i,i)=\frac{1}{n}{2(n-1)\choose n-1},$$

and, for $i>j$, the other nonzero entries satisfy

$$M_n(i,j)=\sum_{\substack{i'\geq i\j'\leq j}}M_{n-1}(i',j').$$

The conjecture extends the known recurrence for the first column and last row, while the complete bigraded Hilbert series of ${\rm DQ}_n$ remains to be established.