Equality conjecture for balanced bipartite biindependence in hypercubes

Let $Q_r$ be the $r$-dimensional hypercube, let $\alpha_{\text{bal}}(Q_r)$ denote the maximum total size of a balanced bipartite biindependent pair in $Q_r$, and let $a(r)$ be the quantity defined in the preceding construction. Equality conjecture. Equality

$$\alpha_{\text{bal}}(Q_r)=a(r-1)$$

holds for all $r\geq 1$. The preceding construction establishes the lower bound $\alpha_{\text{bal}}(Q_r)\geq a(r-1)$; the conjecture asserts that this bound is always tight.