Necessity of the logarithmic factor in the Bobkov–Chistyakov–Götze CLT bound

Determine whether the logarithmic factor log n is necessary in the bound of Theorem 1 (Bobkov–Chistyakov–Götze, Prop. 17.5.1), which asserts that for an isotropic random vector X in ℝ^n there exists a set of directions of measure at least 9/10 such that the Kolmogorov distance between the marginal X·θ and a standard Gaussian is at most (C log n / n) · C_P(X).

Background

The theorem links Gaussian approximation of most one-dimensional marginals to the Poincaré constant, with an error involving (log n)/n. Removing the logarithmic factor would sharpen quantitative central limit phenomena for log-concave measures and convex bodies.

Current techniques yield C_P(X) ≲ log n for isotropic log-concave X, which combined with Theorem 1 gives nontrivial CLT rates; it is open whether the logarithmic factor is intrinsic.