Conjectural determinant factorization for the matrix C

Let $n$ be a positive integer and let $C$ be the $(n-1)\times(n-1)$ matrix whose entries are

$$C_{i,j}=\begin{cases} t-(n-2i+2)(n-1-i)-1,&i=j;\\ (i-1)(n-i),&i=j+1;\\ -(n-1-i)(n-i),&i=j-1;\\ 0,&\text{otherwise}. \end{cases}$$

The determinant factorization conjecture. Writing $n=2r$ or $n=2r-1$, respectively,

$$\det C=\begin{cases} (t-n+1)\displaystyle\prod_{i=0}^{r-2}(t-n+1-r(r-1)+i(i+1))^2,&n=2r;\\ (t-n+1)(t-n+1-(r-1)^2)\displaystyle\prod_{i=1}^{r-2}(t-n+1-(r-1)^2+i^2)^2,&n=2r-1. \end{cases}$$

If true, this factorization implies $t_0=\lfloor (n-1)^2/4\rfloor$, supplying the initial range for the recursion defining $h_n(t)$. The source provides no resolution status.