Recurrence conjecture for $(s,s+r)$-core partitions with $d$-distinct parts

For integers $s$, $r$, and $d$ with $1\le r\le d$, let $N_{d,r}(s)$ denote the number of $(s,s+r)$-core partitions with $d$-distinct parts. Recurrence conjecture. The values of $N_{d,r}(s)$ are characterized by

$$N_{d,r}(s)=s\quad\text{for }1\le s\le d,$$ $$N_{d,r}(d+1)=d+r,$$

and

$$N_{d,r}(s)=N_{d,r}(s-1)+N_{d,r}(s-(d+1))$$

for $s\ge d+2$. This conjecture is based on experimental evidence and was verified in the paper for $s<10$ by listing all relevant partitions; a proof of the stated recurrence and initial values remains open.