Let a Mersenne number be $M\_p=2^{p}-1:p\in\text{prime}$

1. Suppose above some lower bound every Mersenne number were prime and every prime number were a Mersenne number.

Can this conjecture be shown contradictory to the prime number theorem, the properties of Mersenne primes, or any other heuristics?
2. Suppose rather than there being some lower bound, suppose the above conditions represent the limiting behaviour of the primes (i.e. the proportion of primes greater than $n$ that are Mersenne tends to 1 as $n\to\infty$). Does this contradict currently known asymptotics?
3. What do heuristics show about (2)?

---

I'm not great at working out Big O notation etc. I'm aware this question would require the density of primes greater than some $n$ to approach zero so I conclude question 1 is answered "YES, it would be contradictory." But I don't know how the rate at which the density approaches zero would affect the answer to question 2.