This is a problem from my professor, who claimed that it's open:

Combinatorial problem.

Fill $1,2,...,mn$ into a rectangle of size $m\times n$, such that for every number other than $mn$, there is a larger number which is in the same row or column.

Prove there are $\frac{(mn)!m!n!}{(m+n-1)!}$ ways to fill.

It is said that the problem is from a high school student.

I would like to know:

Is the problem really open?

If it's open, are there any references? Is it from a high school student, as claimed?

There's no need to answer the problem to any extent.