Cube-cover conjecture for diagonalization games

Let $n$ and $m$ satisfy $2^{n-1}+1\leq m<2^n$, and set

$$d=2^n-m<2^{n-1}.$$

For a nonempty subset $J\subseteq[n]$, a $J$-cube is the set of binary vectors obtained by fixing all coordinates outside $J$ and allowing the coordinates in $J$ to vary. Cube-cover conjecture. For any collection $J_1,J_2,\ldots,J_q$ of nonempty subsets of $[n]$ satisfying

$$\sum_{i=1}^q |J_i|=d,$$

there are cubes $C_1,\ldots,C_q$ such that $C_i$ is a $J_i$-cube and

$$\left|\bigcup_{i=1}^{q} C_i\right|\geq d+q.$$

This condition is equivalent to Kronecker having a winning strategy when Cantor uses $mn-(2^n-m)$ queries. The statement concerns covering the Boolean cube under prescribed coordinate freedoms and would establish optimality of the proposed strategy; its resolution is not given in the supplied text.