Supersaturation conjecture for even-town families

For a set family $[?]$ and distinct sets $A,B\begin{pmatrix}\in\mathcal{A}\end{pmatrix}$, let $peratorname{op}[?]([?])$ denote the number of pairs whose intersection has odd size. Here $2^{[n]}$ is the power set of $[n]=\{1,\ldots,n\}$, and $[?]$ consists only of even-sized subsets.

Supersaturation conjecture. Let $n\geq 1$ and fix $3\leq s\leq 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}$. If $[?]\subset 2^{[n]}$ consists of even-sized subsets with $|[?]|\geq 2^{\lfloor n/2\rfloor}+s$, then

$$\operatorname{op}(\mathcal{A})\geq s\cdot 2^{\lfloor n/2\rfloor-1}.$$

The statement extends the paper's proved cases $s=1,2$ and asserts that the described even-town construction remains extremal throughout the indicated range; its status is presented as conjectural in the source.