Determinant evaluation conjecture for the matrix T_{n,m}(x)

Let $T_{n,m}(x)$ be the $n\times n$ matrix

$$T_{n,m}(x):=\left[\binom{x+m}{j-i+m}-\binom{x+m}{m-i-j-1}\right]_{i,j=0}^{n-1}.$$

Determinant evaluation conjecture. The determinant satisfies

$$\det T_{n,m}(x)=\prod_{i=1}^n\prod_{j=1}^m\frac{(x+i-j)(x+2i+j-2)}{(x+2i-j)(i+j-1)}.$$

This evaluation is a conjectured closed form for a parameterized determinant arising from a MathOverflow question about a related binomial determinant. The supplied text gives no resolution status.