The level-two weighted sum formula conjecture

For $r\geq 1$ and $0\leq a\leq r$, consider all indices $(k_1,\ldots,k_r)$ with $k_i\in\{1,2\}$ and exactly $a$ entries equal to $2$. Let $\#\{i\mid i\text{ odd},\ k_i=2\}$ count the positions of the entries equal to $2$ that have odd index. Level-two weighted sum formula conjecture.

$$\sum_{\substack{k_i\in\{1,2\}\\ \#\{i\mid k_i=2\}=a}}\left((-1)^{\#\{i\mid i\text{ odd},\ k_i=2\}}2^a-1\right)\zeta_{\mathcal{A}}^{(2)}(k_1,\ldots,k_r)=0.$$

The weight of every term is $r+a$. The conjecture is described as a variant of the weighted sum formula, although the source notes that its form is unusual; no resolution is given.