The cube decomposition conjecture for topological Hochschild homology of Johnson–Wilson spectra

Let $p$ be an odd prime, and let $E(n)$ be a sufficiently commutative $S$-algebra. Write $L_iE(n)$ for the height-$i$ local piece of $E(n)$, and let $dt_1,\dots,dt_{n-1}$ denote the indicated exterior generators in $K(0)_*\mathrm{THH}(E(n))$. For $0\leqslant i\leqslant n-1$, consider monomial generators

$$\omega\in\Lambda_\mathbb{Q}(dt_1,\dots,dt_{n-i-1})\{dt_{n-i}\}\subset K(0)_*\mathrm{THH}(E(n)).$$

The cube decomposition conjecture. The spectrum $\mathrm{THH}(E(n))$ decomposes as a sum of $2^n$ factors: one copy of $E(n)$, together with $2^{n-i-1}$ suspended copies of $L_iE(n)$ for each $0\leqslant i\leqslant n-1$. More precisely, the summand indexed by a monomial $\omega$ is

$$\Sigma^{|\omega|}L_iE(n).$$

This gives the predicted $n$-dimensional cube of local pieces, with the indexing monomials recording the coordinates of the summands. The claim generalizes the displayed decomposition for $\mathrm{THH}(E(2))$; its status is left open in the source.