Milnor's conjecture on the rational independence of Hurwitz zeta values

Let $q$ and $k$ be positive integers greater than one. Consider the Hurwitz zeta values $\zeta(k,a/q)$ for integers $a$ satisfying $1\le a\le q$ and $(a,q)=1$. Milnor's conjecture. These values are linearly independent over $\mathbb{Q}$.

This is a rational linear-independence conjecture for special values of Dirichlet series. The source explains that Milnor generalized earlier nonvanishing questions when the period is not prime, but the material provided does not state whether the conjecture has been resolved.