Somehow $\tau=1.2516475977905$ appears to have the property that $$ \left\lfloor 2^{2^{{\,}^{\cdot^{\cdot^{\cdot^{\tau}}}}}}\right\rfloor $$ is always a prime. Here $\lfloor x\rfloor$ denotes the floor function that returns the integral part of $x$.
Does this hold? If not, what is wrong with it. If yes, then why.