Full-dimension conjecture for the generalized Stieltjes constant span

Let $q>1$ be any positive algebraic number and let $b\geq 3$ be any integer. Let $\mathbb{F}$ be a number field such that $\mathbb{F}$ is linearly disjoint from the cyclotomic field $\mathbb{Q}(\zeta_b)$. Define $V_{\mathbb{F}}(q,b)$ to be the $\mathbb{F}$-linear span of

$$\left\{1,\gamma_0\left(q,\frac{a}{b}\right):1\leq a<b,\ (a,b)=1\right\}.$$

Full-dimension conjecture. One has

$$\operatorname{dim}_{\mathbb{F}}V_{\mathbb{F}}(q,b)=\varphi(b)+1.$$

This is stated as an equivalent formulation of the number-field linear independence conjecture. The supplied text gives no resolution status.