Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$ is prime?

Otherwise, is there any conjecture about that? Can anybody provide any reference for such conjecture in literature?

Thanks a lot!