Broadhurst–Mellit conjecture relating a 7-Bessel moment to a zeta function

Let $\operatorname{\mathbf{IKM}}(2,5;1)$ denote the indicated 7-Bessel moment, and let $\zeta_{7,1}(s)$ be the Euler product

$$\zeta_{7,1}(s)=\prod_p\frac{1}{Z_7(p,p^{-s})}.$$

Broadhurst–Mellit conjecture. One has

$$\operatorname{\mathbf{IKM}}(2,5;1)=\frac{5\pi^2}{24}\zeta_{7,1}(2).$$

The relation is a numerical connection discovered by David Broadhurst working with Anton Mellit. The supplied text says that it still awaits a proof.