The periodicity conjecture for the last non-zero digit sequence

Let $b\geq 2$ be an integer, and let $\mathfrak{S}_b(n)$ denote the last non-zero digit of $n^n$ in base $b$. Periodicity conjecture. The sequence $\mathfrak{S}_b(n)$ is eventually periodic if and only if

$$b=2^{2^s}$$

for some $s\in\mathbb{N}$; moreover, in that case the sequence is periodic. The preceding theorem establishes this assertion for prime-power bases, while the authors report computational evidence that no other bases yield eventual periodicity; the general case remains open.