Connections between hyperlinearity, stability and character rigidity for higher rank lattices

Let $\Gamma$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $\Gamma$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $\Gamma$ is (flexibly) Hilbert--Schmidt stable then: $(a)$ infinite central extensions $\widetilde{\Gamma}$ of $\Gamma$ are not hyperlinear, and $(b)$ every character of $\Gamma$ is either finite-dimensional or induced from the center (character rigidity). As a consequence, a positive answer to the following question would yield an explicit example of a non-hyperlinear group: If two representations of the modular group $\mathrm{SL}_2(\mathbb{Z})$ almost agree on a specific congruence subgroup $H$ under a commensuration, must they be close to representations that genuinely agree on $H$?