Conjecture on vanishing values of seaweed composition counts

Let $C_{p,q}(n,k)$ denote the composition-counting function studied in the paper. Vanishing conjecture.

$$C_{1,4}(2k+1+s,k+s)=0$$

for integers $s,p,k\geq 0$ with $k=2^p$. The values of $n_{k,4}$ suggest this vanishing pattern, but no proof or resolution is given here.