Pfaffian evaluations for Motzkin, Delannoy, Schröder and Narayana sequences

Let $M_n=\sum_{k=0}^n\binom{n}{2k}C_k$ be the Motzkin numbers, $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ the central Delannoy numbers, $S_n=\sum_{k=0}^n\binom{n+k}{2k}C_k$ the Schröder numbers, and $N_n(a)=\sum_{k=0}^n\frac1n\binom{n}{k}\binom{n}{k-1}a^k$ the Narayana polynomials, with $N_0(a)=1$. Motzkin–Delannoy–Schröder–Narayana Pfaffian conjecture. For every integer $n\geq1$,

$$\mathop{\rm Pf}\biggl((j-i)M_{i+j-3}\biggr)_{1\leq i,j\leq 2n}=\prod_{k=0}^{n-1}(4k+1),$$ $$\mathop{\rm Pf}\biggl((j-i)D_{i+j-3}\biggr)_{1\leq i,j\leq 2n}=2^{n^2-1}(2n-1)\prod_{k=1}^{n-1}(4k-1),$$ $$\mathop{\rm Pf}\biggl((j-i)S_{i+j-2}\biggr)_{1\leq i,j\leq 2n}=2^{n^2}\prod_{k=0}^{n-1}(4k+1),$$

and

$$\mathop{\rm Pf}\biggl((j-i)N_{i+j-2}(a)\biggr)_{1\leq i,j\leq 2n}=a^{n^2}\prod_{k=0}^{n-1}(4k+1).$$

These are conjectural Pfaffian evaluations based on lattice-path sequences and computer experiments. Since $C_n=N_n(1)$, $S_n=N_n(2)$, and the stated relation expresses $M_{n-1}$ through $N_n$, proving the Narayana identity would imply the Motzkin and Schröder identities; the conjecture remains open.