Sahin's extended recurrence conjecture for simultaneous core partitions

For positive integers $d$, $r$, and $s$, let $N_{d,r}(s)$ denote the extension, for $s\leq0$, of the number of $(s,s+r)$-core partitions with $d$-distinct parts, defined through the corresponding subsets described in the source. For $r\leq d$, this extension satisfies

Sahin's extended conjecture. For any positive integers $r\leq d$, we have

$$N_{d,r}(s)= \begin{cases} 1, &s=1\\ N_{d,r}(s-1)+N_{d,r}(s-d-1), &s\geq2. \end{cases}$$

This is the reformulation of Sahin's recurrence after extending $N_{d,r}(s)$ to nonpositive $s$; the source proves the analogous reformulation for $r=1$ but gives no resolution for general $r\leq d$.