The $1/3 + \varepsilon$ is not a barrier anymore!

Resorting to estimates of exponential sums over subgroups due to Shrkedov and Shteinikov, J. Cilleruelo and M. Garaev proved in their paper ["On the congruence $x^{x} \equiv 1 \pmod{p}$" (PAMS, **144** (2016), no. 6, pp. 2411 - 2418)][1] the following result:

> Let $J(p)$ denote the number of solutions to the congruence $$x^{x} \equiv 1 \pmod{p}, \quad 1 \leq x \leq p-1.$$ Then, for any $\varepsilon>0$, there exists $c:=c(\varepsilon)>0$ such that $J(p) < c \, p^{27/82 + \varepsilon}$.

This theorem may well be the state of the art on this problem...

  [1]: https://www.ams.org/journals/proc/2016-144-06/S0002-9939-2015-12919-X/S0002-9939-2015-12919-X.pdf