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?
Dogon et al. (Wed,) studied this question.