On universality of countable powers of absolute retracts

We construct an absolute retract X of arbitrarily high Borel complexity such that the countable power X ^ω is not universal for the Borelian class A ₁ of sigma-compact spaces, and the product X ^ω x ∑, where ∑ is the radial interior of the Hilbert cube, is not universal for the Borelian class A ₂ of absolute G _δσ-spaces.