The critical exponent conjecture for powers of doubly nonnegative matrices

Let $m(n)$ denote the smallest real number such that, for every $n$-by-$n$ doubly nonnegative matrix $A$, the conventional power $A^\alpha$ is doubly nonnegative for all $\alpha\geq m(n)$. The quantity $m(n)$ is called the critical exponent.

Critical exponent conjecture. For all integers $n\geq 2$,

$$m(n)=n-2.$$

The conjecture extends the result known for $n<6$, while the general case remains open in the supplied source context.