Minimal-weight-vector conjecture for tight irreducible subcube partitions

A subcube partition $F$ of length $n$ has weight vector $w(F)=(w_0,\ldots,w_n)$, where $w_h$ is the number of subcubes containing $h$ entries equal to $1$. For $h\leq n$, write $w_{\geq h}=w_h+\cdots+w_n$. A weight vector majorizes another if all corresponding upper-tail sums are at least as large. The 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.

Minimal-weight-vector conjecture. For every $n\geq 3$, the minimal weight vectors, under majorization, of tight irreducible subcube partitions of length $n$ are

$$(1,n-1,n-1,0,\ldots,0)$$

and

$$(1,n,n-3,1,0,\ldots,0).$$

These two vectors are realized by explicit constructions in the paper; whether they are the only minimal vectors in general remains open.