The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than $n-2$ and we conjecture that it is always $n-2$ (as it is with Hadamard powering). We prove this conjecture when $n<6$ and in certain other special cases. We establish a quadratic bound for the critical exponent in general.