O'Neill and Versträete's uniform extremal-family conjecture

Let $\binom{[n]}k$ be the family of $k$-element subsets of $[n]$. For $r\geq2$, let ${\rm BD}_r(n,k)$ and $\mathcal F_r(n,k,r-1)$ be the two specified non-trivial $r$-wise intersecting $k$-uniform families. A family is extremal if it has maximum size among non-trivial $r$-wise intersecting subfamilies of $\binom{[n]}k$.

O'Neill and Versträete's conjecture. Let $k>r\geq2$ and $n\geq kr(r-1)$. Then the unique extremal non-trivial $r$-wise intersecting families in $\binom{[n]}k$ are ${\rm BD}_r(n,k)$ and $\mathcal F_r(n,k,r-1)$, up to isomorphism.

This is the proposed uniform analogue of the extremal result for product-measure families. The supplied text gives no resolution of the conjecture.