Xin's second determinant factorization conjecture

Let $cmu$ be an indeterminate, let $n$ be an odd non-negative integer, and use $cdelta_{ij}$ and shifted factorial notation as in the first conjecture.

Xin's second determinant conjecture.

$$cdet_{0cle i,jcle n-1}cleft(-cdelta_{ij}+cbinom{cmu+i+j}{j+2}cright)$$

is equal to the product displayed in the source statement.

The claim concerns the next determinant in the same family; the source says that the factorization occurs for odd $n$ but not for even $n$. Its status is not resolved in the supplied text.