Number-field linear independence conjecture for generalized Stieltjes constants

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 $\mathbb{Q}(\zeta_b)$. The numbers

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

are $\varphi(b)+1$ real numbers.

Number-field linear independence conjecture. The numbers

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

are linearly independent over $\mathbb{F}$.

This is presented as a number-field extension of the conjecture attributed to Chatterjee and Garg. The supplied text does not state whether it has been proved or disproved.