Hajac–Tobolski nilpotent matrix extremal-product conjecture

Let $N$ be a non-negative real and $k$ a positive integer. For a square matrix $A$ with non-negative real entries, define its weight by

$$|A|:=\sum_{i,j}A_{ij}.$$

Assume that $A$ is nilpotent and has weight $N$. Hajac–Tobolski's extremal-product conjecture. The following three quantities are equal: the maximal weight of $A^k$, the maximal product of $k$ non-negative reals with sum $N$, and

$$\left(\frac Nk\right)^k.$$

This is the product formulation of the same nilpotent-matrix extremal claim stated immediately earlier in the source, so it does not add a distinct mathematical assertion; it connects the matrix problem with the balanced-product extremum.