Equivalent odd-subspace formulation of the Chowla–Milnor conjecture

Let $k\geqslant 2$ and $q\geqslant 3$ be integers. For $1\leqslant a<q$ with $\gcd(a,q)=1$, define

$$\zeta^{+}\left(k,\frac{a}{q}\right):=\zeta\left(k,\frac{a}{q}\right)+(-1)^k\zeta\left(k,1-\frac{a}{q}\right),$$ $$\zeta^{-}\left(k,\frac{a}{q}\right):=\zeta\left(k,\frac{a}{q}\right)-(-1)^k\zeta\left(k,1-\frac{a}{q}\right).$$

Define

$$V_k^{-}(q):=\operatorname{Span}_{\mathbb{Q}}\left\{\zeta^{-}\left(k,\frac{a}{q}\right)\mid 1\leqslant a<\frac q2,\ \gcd(a,q)=1\right\}$$

and

$$V_k^{+}(q):=\operatorname{Span}_{\mathbb{Q}}\left\{\zeta^{+}\left(k,\frac{a}{q}\right)\mid 1\leqslant a<\frac q2,\ \gcd(a,q)=1\right\}.$$

Equivalent Chowla–Milnor formulation. One has

$$\dim_{\mathbb{Q}}V_k^{-}(q)=\frac{\varphi(q)}2$$

and

$$V_k^{+}(q)\cap V_k^{-}(q)=\{0\}.$$

Since the even subspace is known to have dimension $\varphi(q)/2$ and $V_k(q)=V_k^{+}(q)+V_k^{-}(q)$, these conditions are equivalent to the Chowla–Milnor linear independence conjecture. The source says that, apart from the case $k=2$, $q=3$, other cases remain open.