Do there exist odd primes $p,q$ with $2^{pq}+1\equiv(2^p+1)^q\pmod{\frac{3^p-1}2}$ and vice versa?

Question

This question is inspired by my partial answer to this question. I would like to know whether there exist any odd primes $p,q$ such that: $$2^{pq}+1\equiv(2^p+1)^q\pmod{\frac{3^p-1}2}$$ $$2^{pq}+1\equiv (2^q+1)^p\pmod{\frac{3^q-1}{2}}$$ So far, I've found no examples with $p,q\le 100$.