Towards the finiteness of \(\pi _*L_{K(n)}S^0\) (Q952415)

Let \(X\) be any spectrum and \(L_{K(n)}X\) be the localization of \(X\) with respect to the \(n\)th Morava \(K\)-theory, \(K(n)\). It is known that \(L_{K(n)}X\) is local with respect to the \(\text{mod}(p)\) Moore spectrum so that the homotopy groups \(\pi_*L_K(n)X\) are canonically a module over the \(p\)-adic integers \(\mathbb Z_p\). This paper is an attempt to make some progress towards the long standing open question asking whether \(\pi_*L_{K(n)}S^0\) is a module of finite type over \(\mathbb Z_p\) for \(n\geq 2\).