Conjecture on the irrationality exponent of the first odd zeta constant

Let $\alpha$ be the unique irrational real number such that

$$\zeta(3)=\alpha\pi^3.$$

Irrationality-exponent conjecture. The irrationality exponent of the first odd zeta constant satisfies

$$\mu(\zeta(3))=\mu(\alpha\pi^3)=2.$$

This conjecture concerns whether the irrationality exponent of $\zeta(3)$, whose best known upper bound has been substantially improved, attains the minimal possible value for an irrational number. The statement also asserts the corresponding equality for $\alpha\pi^3$, using the known relation $\zeta(3)=\alpha\pi^3$ with $\alpha$ irrational.