Conjectural computation of topological Hochschild homology of truncated Brown–Peterson spectra

Let $B\langle n\rangle$ be an $E_3$ form of $BP\langle n\rangle$. For $1\leq m\leq n$ and $n\geq 1$, write $P(-)$ for a polynomial algebra and $E(-)$ for an exterior algebra. Define

$$T_m^n=P(v_m)\otimes E(\lambda_1,\dots,\lambda_{n-m})\{a_{s,\ell}^{(\underline{k})}:\ell\not\equiv p-1\pmod p,s\geq 1\}/\simeq,$$

where $\underline{k}=(\epsilon_1,\dots,\epsilon_m)$ with each $\epsilon_i\in\{0,1\}$, and where $\simeq$ is generated by

$$v_m^{r_n(s,m)}\lambda_1^{\epsilon'_1}\cdots\lambda_{n-m}^{\epsilon'_{n-m}}a_{s,\ell}^{(\underline{k})}\sim 0.$$

Here

$$r_n(s,m)=p^{n-m+s}+p^{n-m+s-(m+1)}+\cdots+p^{n+j-m},$$

with $j\in\{1,\dots,m+1\}$ uniquely determined by $s\equiv j\pmod{m+1}$, and the classes $\lambda_s$ are recursively defined by

$$\lambda_s:=\begin{cases}\lambda_s&1\leq s\leq n+1,\lambda_{s-(m+1)}\mu_{n+1}^{p^{s-(n+2)}(p-1)}&s>n+1.\end{cases}$$

The element $a_{s,\ell}^{(\underline{k})}$ projects to

$$\lambda_{n-m+s}\lambda_{n-m+s+1}^{\epsilon_1}\cdots\lambda_{n+s}^{\epsilon_m}\mu_{n+1}^{\ell p^{s-1}}\in E_\infty^{*,0}.$$

Conjectural THH computation. There is an isomorphism

$$\operatorname{THH}_*(B\langle n\rangle;k(m))\cong P(v_m)\otimes E(\lambda_1,\dots,\lambda_{n-m})\oplus T_m^n.$$

This proposes a uniform description of the topological Hochschild homology of truncated Brown–Peterson spectra with coefficients in $k(m)$, extending the calculations preceding it; the supplied text does not state a resolution or provide evidence that settles the conjecture.