Conjecture: A family of formula for the Euler Constant using primes

Let $p$ be a prime. Experimental data shows that for any $1 \le m < p$,

$$ \lim_{p \to \infty} \left(\sum_{\substack{1\le a,b<p\\ab\equiv m\pmod p}} \frac{1}{b}\left(\frac{b}{a}\right)^{1/p} - \log p \right) = \gamma. $$

The result is not true if $p$ is composite. Can this be proved for disproved?

Related question.