Generic exponential lower-bound conjecture for the Diophantine parameter of random isometries

Let $R_{\mathrm{DPV}}(n,\theta)$ denote the parameter appearing in the Diophantine condition $\mathrm{DPV}(n,R)$ for an angle $\theta\in[0,2\pi)$. The paper proves, when $\cos\theta\in\mathbb{Q}\setminus\frac12\mathbb{Z}$, that $R_{\mathrm{DPV}}(n,\theta)\geq\exp(cn/(\log n)^2)$ for all sufficiently large $n$. Generic exponential lower-bound conjecture. For generic $\theta\in[0,2\pi)$, there exists $c>0$ such that for all sufficiently large $n$,

$$R_{\mathrm{DPV}}(n,\theta)\geq\exp(cn).$$

This conjecture would strengthen the exponential-type lower bound established in the stated rational-cosine case and is intended to capture the generic growth of the Diophantine parameter. The supplied text gives no resolution, so the conjecture remains open.