Projective-plane extremal conjecture for cross-intersecting hypergraphs

For integers $r,m$, let $g(r,m)$ be the maximum, over all $m$-tuples $(H_1,\ldots,H_m)$ of pairwise cross-intersecting $r$-uniform hypergraphs, of

$$\min_{1\leqslant i\leqslant m}\tau^*(H_i),$$

where $\tau^*$ denotes fractional covering number. A projective plane of uniformity $r$ gives the lower bound $g(r,m)\geqslant r-1+1/r$ whenever such a plane exists.

Projective-plane extremal conjecture. For $m\geqslant r+2$,

$$g(r,m)\leqslant r-1+\frac1r,$$

with equality if and only if there is a projective plane of uniformity $r$. The conjecture is proved in the source for $r=2$, and the paper proves that the bound holds for every $r$ once $m$ is sufficiently large; the general assertion remains open.