Minimax volume conjecture for Steenrod-square cup powers on rectangles

Let $R$ be an $n$-dimensional rectangle with side lengths $R_1\leq\cdots\leq R_n$. Let $\mathbb{V}_R$ denote the minimax volume, let $a(k,R)$ be the relevant cup-power cohomology class, and let $Sq_{n-l}^Q$ denote the indicated Steenrod-square operation with parameter $Q$. For each integer $k+1\leq l\leq n$, the conjecture is

Minimax volume conjecture. The minimax volume satisfies, up to a constant factor depending only on $n$,

$$\mathbb{V}_R\bigl(Sq_{n-l}^Q a(k,R)\bigr)\sim \inf_{0\leq j\leq k} R_1\cdots R_j\bigl(R_{j+1}\cdots R_l\bigr)^{\frac{k-j}{l-j}}2^{\frac{l-k}{l-j}Q}.$$

Such an estimate would provide lower bounds for the $k$-dilation of degree-one maps between rectangles and, in particular, would imply the stated rectangular lower bound. The source presents this as a conjectural estimate; no resolution is given here.