A binomial identity for valid compositions

Let $n$ and $k$ be nonnegative integers, and let $C_q$ and $D_q$ be the quantities defined in Theorem 8 for compositions $q$. Write $\operatorname{Comp}_{k+1}(n-k)$ for the set of compositions of $n-k$ with $k+1$ parts. The binomial-sum conjecture. In the notation of Theorem 8,

$$\sum_{q\in\operatorname{Comp}_{k+1}(n-k)}C_qD_q=\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}$$

for all nonnegative integers $n$ and $k$. Numerical evidence suggests this identity, which would give a term-by-term refinement of the generating-function identity obtained by combining Theorems 8 and 9.