Does there exist a real number $c > 1$ such that for every natural number $n > 0$, the number $\lfloor c^n \rfloor$ is prime?

I doubt such a number $c$ is known to exist, since the best similar results I've seen are much weaker. For example, in 1947 William Harold Mills proved that there is a real number $c > 1$ such that for every natural number $n > 0$, the number $\lfloor c^{3^n} \rfloor$ is prime:

• Wikipedia, Mills' constant

and if the Riemann Hypothesis holds, the smallest such $c$ is approximately

$$ c = 1.3063778838630806904686144926\ldots $$

Harder results along these lines are known, e.g. in 2010 Matomäki showed that there exists an uncountable infinity of real numbers $c > 1$ with the property that for every natural $n > 0$, the number $\lfloor c^{2^n} \rfloor$ is prime:

• K. Matomäki, Prime-representing functions, Acta Mathematica Hungarica 128 (4) (2010), 307–314.

However, I haven't seen results like this for $\lfloor c^n \rfloor$. So, I think my question boils down to: is the answer to my original question known to be no, or is it still open?