Supersaturation conjecture for odd-town families

Let $[n]$ be an $n$-element ground set, and let $[?]$ be a collection of odd-sized subsets of it. Write $peratorname{op}[?]([?])$ for the number of distinct pairs of sets in $[?]$ having odd-sized intersection.

Odd-town supersaturation conjecture. Let $n\geq 1$ and fix $1\leq s\leq n$. If $[?]$ is a collection of odd-sized subsets of an $n$-element set with $|[?]|\geq n+s$, then

$$\operatorname{op}(\mathcal{A})\geq 3s.$$

This extends the proved case $s=1$, for which every odd-sized family of size at least $n+1$ has at least three odd-intersection pairs; the conjecture predicts the extremality of the construction obtained by adjoining $s$ triples from vertex-disjoint copies of $K_4^{(3)}$.