Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all } 2 \text{-generated groups of order less than }n}{\text{the number of all groups of order less than } n} = 0$?

This statement is implied by a well known conjecture:

$\lim_{n \to \infty} \frac{\text{the number of all groups of nilpotency class } 2 \text{, exponent } 4 \text{ and order less than }n}{\text{the number of all groups of order less than } n} = 1$

(because the $2$-generated relatively free group of nilpotency class $2$ and exponent $4$ is finite)

However, the problem with aforementioned statement is, that it is...

...well, just a conjecture.

Is there a way to strictly prove the main statement of the question?