The diagonal maximal-covering-number conjecture for cross-intersecting families

Let $m(k,k)$ denote the maximum product of the sizes of two cross-intersecting families with maximal covering number $k$ in the setting of the paper. The diagonal conjecture. For all $k\geq 2$,

$$m(k,k)=(k^{k-1}+k-1)^2.$$

The authors note that this agrees with the proved cases $m(2,2)=3^2$ and $m(3,3)=11^2$, and with the construction giving the bound for large values of $k$; the assertion remains open in general.