  We prove an extension of Szarek's optimal Khinchin inequality (1976) for
distributions close to the Rademacher one, when all the weights are uniformly
bounded by a $1/\sqrt2$ fraction of their total $\ell_2$-mass. We also show a
similar extension of the probabilistic formulation of Ball's cube slicing
inequality (1986). These results establish the distributional stability of
these optimal Khinchin-type inequalities. The underpinning to such estimates is
the Fourier-analytic approach going back to Haagerup (1981).
