Conjectured first diagonal formula for $C_{2,t}$

Let $C_{2,t}(n,k)$ be the composition-counting function. First-diagonal conjecture. If $t$ is even, then

$$C_{2,t}(t+s,\tfrac{t}{2}+s)=\tfrac{t}{2}+\tfrac{t}{2}s$$

for integers $t>2$ and $s\geq 0$. This describes one of the first nonzero diagonals in the even-$t$ case; no proof or resolution is given.