Polynomial product formula conjecture for the generalized Aztec-diamond matrix

Let $A(k,t)$ be the skew-symmetric matrix whose entries satisfy

$$\begin{cases} a_{1,j}=t, & j>1,\\ a_{i,j}=a_{i-1,j}+a_{i,j-1}+a_{i-1,j-1}, & j>i+1,\quad i>1,\\ a_{i,j}=a_{i-1,j}+a_{i-1,j-1}+k(-1)^{i-1}, & j=i+1,\quad i>1,\\ a_{i,j}=-a_{j,i}, & j\leq i. \end{cases}$$

Write $A_{[2n]}(k,t)$ for its leading $2n\times 2n$ principal submatrix, and let $\mathsf{pf}$ denote the Pfaffian.

Generalized product formula conjecture. There exists a sequence of polynomials $o_n(k,t)\in\mathbb{Z}[k,t]$ such that

$$\mathsf{pf}(A_{[2n]}(k,t))=t\,o_{n-1}(k,t)o_n(k,t).$$

The conjecture has been checked by computer up to $n=25$. Setting $k=t=2$ recovers the preceding product formula conjecture, while a proof for general $k$ and $t$ remains open.