Polynomial growth of 2^n σ_n

Determine whether the quantity 2^n σ_n, where σ_n denotes the supremum of densities of lattice sphere packings in ℝ^n, grows at most polynomially in n.

Background

Let σ_n be the supremum of densities of lattice sphere packings in ℝ^n. Classic lower bounds due to Rogers and subsequent refinements suggested growth of order n·2^{-n} up to logarithmic factors, and this paper improves the lower bound to c n^2·2^{-n}. Upper bounds are exponentially small, leaving a substantial gap.

Venkatesh proposed a conjecture on the scaled quantity 2^n σ_n, predicting that it grows at most polynomially in n. Confirming or refuting this would significantly clarify the asymptotic behavior of optimal lattice packing densities in high dimensions.