O'Neill's supersaturation conjecture for eventown families

Let $n>1$, let $[n]=\{1,2,\ldots,n\}$, and let $2^{[n]}$ denote the family of all subsets of $[n]$. For a family $\mathcal{A}\subset 2^{[n]}$, write $op(\mathcal{A})$ for the number of pairs of distinct members $A,B\in\mathcal{A}$ such that $|A\cap B|$ is odd. Fix an integer $s$ satisfying

$$1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}.$$

O'Neill's supersaturation conjecture. If $\mathcal{A}\subset 2^{[n]}$ consists of even-sized subsets and

$$|\mathcal{A}|\ge 2^{\lfloor n/2\rfloor}+s,$$

then

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

This conjecture gives a supersaturation bound for eventown: once an even-set family exceeds the extremal size $2^{\lfloor n/2\rfloor}$, it predicts a linear lower bound on the number of pairs whose intersection has odd size. The source attributes the conjecture to O'Neill; the supplied material does not establish whether it has been resolved.