Rigidity of Fibonacci representations of mapping class groups

We prove that level $5$ Witten-Reshetikhin-Turaev $\mathrm{SO}(3)$ quantum representations, also known as the Fibonacci representations, of mapping class groups are locally rigid. More generally, for any prime level $\ell$, we prove that the level $\ell$ $\mathrm{SO}(3)$ quantum representations are locally rigid on all surfaces of genus $g\geq 3$ if and only if they are locally rigid on surfaces of genus $3$ with at most $3$ boundary components. This reduces local rigidity in prime level $\ell$ to a finite number of cases.