Chowla–Milnor conjecture on linear independence of Hurwitz zeta values

For $s>1$ and $a>0$, let $\zeta(s,a)=\sum_{k\geq 0}\frac{1}{(k+a)^s}$ denote the classical Hurwitz zeta function. For integers $q>1$ and $n>2$, define

$$\mathfrak{L}=\left\{\zeta\left(q,\frac{m}{n}\right):1\leq m\leq n,\ \operatorname{gcd}(m,n)=1\right\}.$$

Chowla–Milnor conjecture. The set $\mathfrak{L}$ is linearly independent over $\mathbb{Q}$. The conjecture is important because its consequences include conditional irrationality results for ratios of odd zeta values to powers of $\pi$, and it also has connections with multiple zeta values. Its general status is not resolved in the supplied text.