Chatterjee–Garg linear independence conjecture for generalized Stieltjes constants

Let $q>1$ be any positive algebraic number and let $b\geq 3$ be any integer. 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.

Chatterjee–Garg 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 the field of rationals.

This conjecture concerns the linear independence of special values of the $q$-analogue of the generalized Stieltjes constants at reduced rational arguments. Its status is not specified in the supplied text.