Invertibility conjecture for the depth-graded sum odd multiple zeta matrix

For $r\geq3$ and $N\geq r+2$ with $N-r\equiv0\pmod 2$, let $T_{N,r}$ be the index set used in the source, and let $E$ be the matrix whose rows and columns are indexed by $T_{N,r}$, with entries $e\binom{k_1,k_2,\ldots,k_r}{n_1,n_2,\ldots,n_r}$ as in the source's proposition.

Invertibility conjecture. For any $r\geq3$, $N\geq r+2$, and $N-r\equiv0\pmod 2$, the order $|T_{N,r}|$ matrix

$$E=\left(e\binom{k_1,k_2,\cdots,k_r}{n_1,n_2,\cdots,n_r}\right)$$

is invertible.

The matrix arises in the explicit calculation of the depth-graded map on sum odd motivic multiple zeta values. The source does not state whether this conjecture has been resolved.