Differentiability conjecture for the arithmetic Fourier series $F_k$ and $G_k$

Let $F_k$ and $G_k$ be the arithmetic Fourier series considered in the paper, and let $q_n(x)$ denote the denominators of the continued-fraction convergents of $x$. Let $k\in\mathbb{N}^*$ be even. Differentiability conjecture. (i) Neither $F_k$ nor $G_k$ is differentiable at any $x\in\mathbb{Q}$; however, $G_k$ is right and left differentiable at each $x\in\mathbb{Q}$. (ii) The function $G_k$ is differentiable at any $x\in\mathbb{R}\setminus\mathbb{Q}$. (iii) The function $F_k$ is differentiable at $x\in\mathbb{R}\setminus\mathbb{Q}$ if and only if

$$\sum_{n=0}^{\infty}\frac{\log q_{n+1}(x)}{q_n(x)^k}<\infty.$$

The conjecture extends the differentiability results proved in the paper for the case $k=2$ to every even positive integer. The preceding discussion indicates that removing the additional conditions used in the $k=2$ theorem is expected to be necessary, while the proposed general statement remains unproved in the source.