Exponential localization and rotational symmetry of INB density-of-states measures

Let $L^{(+,k)}$ and $L^{(\times,k)}$ be the two matrices defined by the additive and multiplicative INB Lax matrices, respectively, both endowed with the probability distribution $\mathrm{d}\mu_{\mathrm{INB}}$ from the INB Gibbs measure. Let $\nu^{\gamma,\beta}_{\mathrm{INB},+,k}$ and $\nu^{\gamma,\beta}_{\mathrm{INB},\times,k}$ denote their densities of states. Define the two hypotrochoids

$$\gamma_{+,k}(t,\eta,\beta)=e^{-it}+\frac{\eta}{\beta}e^{ikt},\qquad \gamma_{\times,k}(t,\eta,\beta)=\frac{\eta}{\beta}e^{-it}+e^{ikt},\qquad t\in[0,2\pi).$$

Exponential localization and symmetry conjecture. The densities of states $\nu^{\gamma,\beta}_{\mathrm{INB},+,k}$ and $\nu^{\gamma,\beta}_{\mathrm{INB},\times,k}$ exist and satisfy the discrete rotational symmetries

$$\nu^{\gamma,\beta}_{\mathrm{INB},+,k}(\mathrm{d}z)=\nu^{\gamma,\beta}_{\mathrm{INB},+,k}\left(e^{\frac{2\pi i}{k+1}}\mathrm{d}z\right),\qquad \nu^{\gamma,\beta}_{\mathrm{INB},\times,k}(\mathrm{d}z)=\nu^{\gamma,\beta}_{\mathrm{INB},\times,k}\left(e^{\frac{2\pi i}{k+1}}\mathrm{d}z\right).$$

Moreover, the densities are exponentially localized in a neighbourhood of $\gamma_{+,k}(t,\eta,\beta)$ and $\gamma_{\times,k}(t,\eta,\beta)$, respectively. The conjecture concerns the limiting spectral distributions of these random Lax matrices and connects their supports with the hypotrochoid curves observed numerically; no resolution is supplied in the source.