Number of maximal subgroups of the Cantor space homeomorphism group

Determine how many maximal subgroups the homeomorphism group of the Cantor space, operatorname{Homeo}(2^N), has.

Background

The paper establishes that several homeomorphism groups, including those of the rational numbers, Baire space, N \times 2^N, \omega^2, and the Sorgenfrey line, have precisely 2^{\mathfrak c} maximal subgroups. The Cantor space 2^N is excluded from the main counting theorem because it is homeoclastic but cannot be partitioned into infinitely many topological moieties, and hence is not \aleph_0-homeoclastic. The authors note that point stabilisers already provide at least \mathfrak c maximal subgroups of \operatorname{Homeo}(2^N), while each point stabiliser is isomorphic to \operatorname{Homeo}(N \times 2^N) and therefore contains 2^{\mathfrak c} maximal subgroups. Nevertheless, the total number of maximal subgroups of \operatorname{Homeo}(2^N) is left unresolved.