Existence of universal topological groups of uncountable weight

Determine whether there exists, for some uncountable cardinal τ, a universal topological group of weight τ; equivalently, establish whether there is a topological group G such that every topological group of weight at most τ embeds topologically as a subgroup of G.

Background

Uspenskiy proved that there exists a universal topological group with a countable base, namely the homeomorphism group of the Hilbert cube with the compact–open topology. This raises the natural extension to uncountable weights.

The paper discusses why straightforward generalizations of Uspenskiy’s approach are blocked: topological characterizations of Tychonoff cubes I^τ for uncountable τ require absolute retract conditions that fail for many compact convex subsets, undermining attempts to use analogous methods to show universality of groups like ℋ(I^τ).