For the Hellinger–Reissner principle of stress-displacement linear elasticity, we prove inf-sup stability bounds which are independent of polynomial degree and mesh size, for a novel sequence of conforming finite element spaces discretizing the stress elasticity complex in two dimensions. The stress space coincides with the symmetry-enforcing Hu–Zhang element, of which we provide a novel characterization. This may be achieved via the construction of a bounded right inverse of the divergence operator, on the discontinuous Lagrange element which discretizes the displacement, by adapting techniques used to proved the analogous statement for the Falk–Neilan pair for incompressible Stokes flow. Our main novelty, however, lies in an alternative proof via the construction of bounded Poincaré operators for the elasticity complex which are polynomial-preserving, in the Bernstein–Gelfand–Gelfand framework of the finite element exterior calculus. Numerical examples are provided.