The Hlawka zeta isometry conjecture for nonconstant radial functions

Let $r\bigl(\theta\bigr)$ be a positive continuous function on the circle, viewed as a radial function, and suppose that infinitely many Fourier coefficients $\hat{r}(n)$ are nonzero. For a linear transformation $g$ and the associated Hlawka zeta function $Z_r(s)$, write $g\cdot r$ for the transformed radial function. Hlawka zeta isometry conjecture. One has

$$Z_r(s)=Z_{g\cdot r}(s)$$

if and only if $g\in O(2,\mathbb{Z})$, namely, if and only if $g$ is an isometry of the lattice $\mathbb{Z}^2$. This concerns the stabilizer of the induced $GL(2,\mathbb{R})$ action on Hlawka zeta functions. The conjecture asserts that, under the stated nondegeneracy condition on $r$, only lattice isometries preserve the zeta function.