Alternating-sum Catalan conjecture for reduced valid hook configurations

For $k\geq 1$, let $C_k=\frac{1}{k+1}{2k\choose k}$ be the $k$th Catalan number, and let $\operatorname{RedVHC}_k(\operatorname{Av}_n(312))$ be the reduced valid hook configurations with $k$ hooks on permutations avoiding $312$.

Alternating-sum Catalan conjecture. For every $k\geq 1$,

$$\sum_{i=1}^k(-1)^{k-i}\left|\operatorname{RedVHC}_k(\operatorname{Av}_{2k+i}(312))\right|=C_k.$$

The identity is one of the additional patterns suggested by the displayed coefficient triangle; the source gives no resolution status.