Sharp balanceability conjecture for the exterior derivative

Let $d$ and $q$ be integers with $1\leq q\leq d-1$, and consider the exterior derivative acting on $(d-q)$-forms. The operator is $k$-balanceable when it satisfies the balanceability property with parameter $k$. Sharp balanceability conjecture. The exterior derivative acting on $(d-q)$-forms is $k$-balanceable if and only if

$$k\in\{q,\dots,d-1\}.$$

The conjecture asserts the sharp range of parameters for which the exterior derivative is balanceable; the surrounding argument establishes sufficiency, while the necessity and hence the exact range remain to be proved.