Frankl–Kupavskii weighted-construction conjecture for the Erdős–Kleitman problem

Let $[n]=\{1,2,\ldots,n\}$, let $e(n,s)$ be the maximum of $|\mathcal F|$ over families $\mathcal F\subseteq 2^{[n]}$ with matching number $\nu(\mathcal F)<s$, and, for a weight function $w\colon[n]\to\mathbb R_{\ge0}$, define

$$\mathcal F(w)=\left\{F\subseteq[n]\colon\sum_{x\in F}w(x)\ge1\right\}.$$

If $\sum_{i=1}^n w(i)<s$, then $\nu(\mathcal F(w))<s$. Frankl–Kupavskii conjecture. For any $n,s$, the maximum in the definition of $e(n,s)$ is attained by a family $\mathcal F(w)$ for some $w\colon[n]\to\mathbb R_{\ge0}$. This meta-conjecture would reduce the Erdős–Kleitman extremal problem to weighted constructions; its resolution is not specified in the source.