Generalized Pfaffian product formulas for Motzkin triangle entries

Let $n$ and $k$ be positive integers. Define the Motzkin-triangle entries by

$$\mathcal{M}^{(k)}_{i}=\left(\!\genfrac{}{}{0pt}{}{i-1}{k-1}\!\right){}_2F_1\!\left(\genfrac{}{}{0pt}{}{\frac{k-i}{2},\frac{k-i+1}{2}}{k+1};4\right).$$

For a $2n\times 2n$ skew-symmetric matrix, let $\operatorname{Pf}$ denote its Pfaffian. Generalized Motzkin Pfaffian conjecture. The Pfaffian

$$\operatorname{Pf}\!\left((j-i)\mathcal{M}^{(k)}_{i+j-2}\right)_{1\leqslant i,j\leqslant 2n}$$

equals

$$\prod_{i=0}^{m-1}\prod_{j=0}^{k-1}(4ki+2j+k)$$

if $m=n/k$ is an integer, and equals

$$\left(\prod_{j=1}^{\lfloor k/2\rfloor}\frac{1}{2j-k}\right)\left(\prod_{i=0}^{m-1}\prod_{j=1}^{k}(4ki+2j-k)\right)$$

if $k$ is odd and $m=(n+\lfloor k/2\rfloor)/k$ is an integer; it is zero in all other cases. Meanwhile,

$$\operatorname{Pf}\!\left((j-i)\left(\mathcal{M}^{(k)}_{i+j-2}+\mathcal{M}^{(k)}_{i+j-1}\right)\right)_{1\leqslant i,j\leqslant 2n}$$

equals

$$\prod_{i=0}^{m-1}\prod_{j=0}^{k-1}(4ki+2j+k+1)$$

if $m=n/k$ is an integer, and equals

$$\left(\prod_{j=1}^{k/2}\frac{1}{2j-k-1}\right)\left(\prod_{i=0}^{m-1}\prod_{j=1}^{k}(4ki+2j-k-1)\right)$$

if $k$ is even and $m=(n+k/2)/k$ is an integer; it is zero in all other cases. The formulas generalize the preceding Pfaffian evaluation for the Motzkin numbers and would provide product evaluations for broad families of Pfaffians built from Motzkin-triangle entries. The source does not provide evidence resolving this proposed generalization.