The DWBC4 Schröder-path enumeration conjecture

For nonnegative integers $a,b,c$, let $N_{a,b,c}$ denote the number of 20V configurations on the rectangular grid of size $(a+b+1)\times(b+c+1)$ with DWBC4-type boundary conditions. Let $S^{(L)}_{u,v}(M)$ denote the partition function of a Schröder path from height $u$ to height $v$, of horizontal extent $M$, constrained to the strip $0\leq y\leq L$. The DWBC4 enumeration conjecture. The two identified families satisfy

$$N_{a,b,1}=S^{(b+1)}_{1,b}(2a+b+1),\qquad N_{0,b,c}=S^{(b)}_{0,b}(2c+b).$$

These equalities assert conjecturally that the indicated DWBC4 configuration counts are enumerated by domino-tiling domains, equivalently by the corresponding single constrained Schröder paths. The source does not establish the equalities in general.