Does the existence of a power of $2$ prime-representing function $\lfloor \alpha^{2^n} \rfloor$ imply Legendre's conjecture to be true?

The Legendre conjecture

for every $n$ there is a prime $ p \in [n^2,(n+1)^2]$

is the same as

for every $a_0 \ge 3$ there exists $\alpha \in [a_0,a_0+1]$ such that $a_n = \lfloor \alpha^{2^n} \rfloor$ is prime for every $n \ge 1$

This is because for every $n$ and for every $k \in [a_n^2,(a_n+1)^2]$, we can refine $\alpha$ without changing $a_0, \ldots, a_n$ to obtain $$a_{n+1} = \lfloor \alpha^{2^n} \rfloor = k$$