Let $p = 2 m {q^2} + 1$ be a prime, where $m$ is a smooth integr and $q$ be any odd prime, and ${g_1},{g_2},h$ be primitive elements of ${\mathbb{F}_p}$ where are related by $${g_1}^{{x_1}} = h\,\,\,,\,\,\,\,{g_2}^{{x_2}} = h,$$ for some unknown integers ${x_1},{x_2}$. We know that $${x_1} \equiv {x_2}\left( {\bmod \,\,{q^2}} \right),$$ Furthermore, these discrete logarithms are available: $$Lo{g_{{g_1}}}{g_2}\,\,\,\,\,\,,\,\,\,\,\,\,\,\,Log{_{g_2}}{g_1}.$$
Can we efficiently recover the original exponents ${x_1}$ or ${x_2}$ based on the given information?