Consider the number $(2m)^{2^n}+1$ where both $m$ and $n$ are positive integers.
Can it be shown that for any given $n$, there exists an $m$ such that $(2m)^{2^n}+1$ is a prime number.
Edit: It can be shown that there exists an integer k such that (2m)2n can be written k2 for any integers m and n. If the conjecture in the question was true, then it would thus imply there are an infinite number of primes of the form k2+1. This is Landau's fourth problem which I believe still remains unsolved. Therefore I suspect this conjecture also cannot be proved at the current time
Showing that for every positive integer $n$ there is a positive integer $m$ , such that $(2m)^{(2^n)}+1$ is prime, would not just solve the $k^2+1$-problem. It would also show that there infinite many primes of the form $k^4+1,k^8+1,k^{16}+1$ and so on.
So, a positive answer is very unlikely. But a negative answer is unlikely as well considering that it is not known whether infinite many Fermat-primes exist.
So, this question will probably remain undecided.