(somewhat related to my earlier question)

Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$).

The equation ${^4}x=2$ has a positive root $x=1.4466014324...$

Is it possible to prove it irrational?