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

Let $Γ$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $Γ$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $Γ$ is (flexibly) Hilbert--Schmidt stable then: $(a)$ infinite central extensions Γ of $Γ$ are not hyperlinear, and $(b)$ every character of $Γ$ 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 SL₂(Z) almost agree on a specific congruence subgroup H under a commensuration, must they be close to representations that genuinely agree on H?