Frankl--Wang conjecture for non-trivial cross-intersecting families of different uniformities

Let $n\ge 2a\ge 2b\ge 4$, and let $A_0\in\binom{[2,n]}{a}$ and $B_0\in\binom{[2,n]}{b}$ satisfy $A_0\cap B_0\ne\emptyset$. Define

$$\mathcal{A}_0=\{A_0\}\cup\{A\in\binom{[n]}{a}:1\in A,\ A\cap B_0\ne\emptyset\},$$ $$\mathcal{B}_0=\{B_0\}\cup\{B\in\binom{[n]}{b}:1\in B,\ B\cap A_0\ne\emptyset\}.$$

Two families are cross-intersecting if every member of one intersects every member of the other, and are non-trivial if neither family is a star. Frankl--Wang's conjecture. If $\mathcal{A}\subset\binom{[n]}{a}$ and $\mathcal{B}\subset\binom{[n]}{b}$ are non-trivial cross-intersecting families, then

$$|\mathcal{A}||\mathcal{B}|\le |\mathcal{A}_0||\mathcal{B}_0|.$$

This proposes the cross-Hilton--Milner-type pair as the extremal construction when the two families have different uniformities. The supplied source does not state whether the conjecture has been resolved.