Rigidity conjecture for random $SL_2(\mathbb{F}_p)$-circulant matrices

Let $p$ be a prime, let $G=SL_2(\mathbb{F}_p)$, and let $f:G\rightarrow\{0,1\}$ be chosen uniformly at random. Write $M_G(f)$ for the corresponding $G$-circulant matrix. Rigidity conjecture. For large primes $p$, the random $G$-circulant $(0,1)$-matrix $M_G(f)$ is Valiant-rigid over $\mathbb{C}$ with high probability. This conjecture proposes rigidity for matrices associated with groups whose irreducible complex representations have large degree, extending the paper’s rigidity questions beyond abelian groups. Its resolution is not supplied here.