The Hlawka zeta orbit conjecture for differentiable radial functions

Let $r$ and $r^{\ast}$ be positive continuously differentiable functions on the circle, and let $Z_r(s)$ denote the Hlawka zeta function associated with $r$. The group $GL(2,\mathbb{R})$ acts on radial functions by $r\mapsto g\cdot r$, and the corresponding orbit consists of all functions obtained from $r$ by this action. Hlawka zeta orbit conjecture. If

$$Z_r(s)=Z_{r^{\ast}}(s),$$

then $r^{\ast}$ belongs to the $GL(2,\mathbb{R})$-orbit of $r$. The conjecture strengthens the orbit-stabilizer question for the induced action of $GL(2,\mathbb{R})$ on Hlawka zeta functions: although the paper gives a counterexample without differentiability, it proposes that continuous differentiability may restore orbit-determination.