Milnor's linear independence conjecture for Hurwitz zeta values

Let $p$ be a prime, let $k>1$ be an integer, and let the Hurwitz zeta function be

$$\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s},$$

for $0<x\le1$. Milnor's conjecture. The real numbers

$$\zeta(k,1/p),\ \zeta(k,2/p),\ \ldots,\ \zeta(k,(p-1)/p)$$

are linearly independent over $\mathbb{Q}$. This is presented as a reformulation of the Chowla–Chowla conjecture. Its status is not specified in the source.