The irrationality-measure conjecture for Catalan's constant

Let

$$\beta(2)=\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)^2}$$

be Catalan's constant, and let $\mu(\beta(2))$ denote its irrationality measure. Irrationality-measure conjecture. The irrationality measure of Catalan's constant is

$$\mu\left(\beta(2)\right)=2.$$

This is an open problem closely related to the Bateman–Horn conjecture; the surrounding discussion connects it with the existence of infinitely many consecutive prime triples of the specified congruence types.