Maximum-minimum-dimension conjecture for tight irreducible subcube partitions

For a subcube partition $F$, let $\delta(F)$ denote the minimum dimension of a subcube in $F$. A subcube partition is tight if every coordinate is constrained by some subcube, and irreducible if no proper subset of more than one subcube has a union that is a subcube.

Maximum-minimum-dimension conjecture. Every tight irreducible subcube partition $F$ of length $n$ satisfies

$$\delta(F)\leq n/2-o(n).$$

The constructions in the paper motivate this asymptotic upper bound, but no general proof is given.