The THH homotopy-ring conjecture for truncated Brown–Peterson algebras

For $n\geq 0$ and $p>2$, let $A_*=\operatorname{BP}\langle n\rangle_*[x]/x^e$. Let $w_{n+1,i}$, $y_{j,i}$, and $\gamma_{p^m}(\sigma^2x^e)$ denote the classes introduced in the preceding construction, with $i\geq 0$ and $j\not\equiv p-1\pmod p$. THH homotopy-ring conjecture. The ring spectrum $\operatorname{THH}(\operatorname{BP}\langle n\rangle[x]/x^e/\operatorname{MU}[x])$ has homotopy ring

$$A_*[w_{n+1,i},y_{j,i}:i\geq 0,\ j\not\equiv p-1\pmod p]\otimes_{A_*}A_*[1/p^m(\sigma^2x^e)^{p^m}:m\geq 0],$$

i.e. the ring map in the preceding remark is a surjection and its kernel is generated by $\{p^m\gamma_1(\sigma^2x^e)-\gamma_{p^m}(\sigma^2x^e)\}$. The source states that this is true for $n=0$; the conjecture concerns the remaining cases with $n\geq 0$ and $p>2$.