On the Hochschild homology of Johnson-Wilson spectra

Let E(n) denote the n-th Johnson-Wilson spectrum at an odd prime p.
The spectrum E(1) coincides with the Adams summand of p-local topological K-theory.
McClure and Staffeldt offered an intriguing computation
of THH(E(1)), showing that it splits as a wedge sum of E(1) and a rationalized suspension of E(1).

In joint work with Birgit Richter, we study the Morava K-theories of THH(E(n)),
with an aim at investigating if McClure-Staffeldt's splitting in lower chromatic
pieces generalizes.  Under the assumption that E(2) is commutative, we show that
THH(E(2)) splits as a wedge sum of E(2) and its lower chromatic localizations.