On closed left ideal decompositions of \(G^{\ast}\) (Q1928423)

It has long been known that for every countably infinite group \(G\) (in particular, for \(G=\mathbb{Z}\)), \(G^*=\beta G \setminus G\) can be decomposed into \(2^c\) left ideals of \(\beta G\). I. Protasov showed that \(G^*\) can be decomposed into \(2^c\) closed left ideals of \(\beta G\) such that the corresponding quotient space of \(G^*\) is Hausdorff. The proof is complicated, based on balleans and slowly oscillating functions. It was simplified by M. Filali, P. Salmi and D. Davenport, N. Hindman and also Y. Zelenyuk separately. For every \(p\in G^*\), \((\beta G)p\) is the principal left ideal of \(\beta G\) generated by \(p\). A principal left ideal of \(\beta G\) is maximal if it is not properly contained in any other principal left ideal of \(\beta G\). If \(G\) is countable, then any two principal left ideals of \(\beta G\) are either disjoint or one of them is contained in the other, and consequently, any two distinct maximal principal left ideals of \(\beta G\) are disjoint. It is an old difficult question whether every point of \(\mathbb{Z}^*\) lies in a maximal principal left ideal of \(\beta \mathbb{Z}\), or equivalently, whether maximal principal left ideals of \(\beta \mathbb{Z}\) form a decomposition (= partition) of \(\mathbb{Z}^*\). For every countably infinite group \(G\), let \(\mathcal{I}(G)\) denote the finest decomposition of \(G^*\) into closed left ideals of \(\beta G\) with the property that the corresponding quotient space of \(G^*\) is Hausdorff. The aim of this paper is to prove the following: Is it consistent with ZFC that, if \(G\) is a countably infinite group which can be embedded algebraically into a compact group, every \(I\in \mathcal{I}(G)\) contains \(2^c\) maximal principal left ideals of \(\beta G\) generated by prime elements?