The conjecture on the error in estimated numbers of discrete logarithm fixed points

Let $p$ be prime, let $g$ and $h$ range over the parameters defining the fixed-point counting function, and let $N_{, g \mathop\mathsf{ANY}, h \mathop\mathsf{ANY}}(p)$ denote the corresponding estimate. For every fixed $\epsilon>0$, consider primes $p\leq x$ for which its deviation from $p-1$ exceeds $p^{1/2+\epsilon}$. The error-distribution conjecture. There are $o(x/\ln x)$ primes $p\leq x$ for which

$$\left|N_{, g \mathop\mathsf{ANY}, h \mathop\mathsf{ANY}}(p)-(p-1)\right|>p^{1/2+\epsilon}$$

for every $\epsilon>0$. This is motivated by an independence heuristic: the expected value of the relevant random model is $p-1$, while its standard deviation is less than $p^{1/2+\epsilon}$ for every $\epsilon>0$ when $p$ is sufficiently large. The statement predicts that only a negligible number of primes have a larger error, but no proof or resolution is supplied here.