The conjectural computation of $\operatorname{TR}^3_{9-\lambda_4}(\mathbb{Z};2)$

Let $\operatorname{TR}^3_{9-\lambda_4}(\mathbb{Z};2)$ be the indicated equivariant topological Hochschild homology group, where $\lambda_4$ is the relevant $RO(S^1)$-representation. The $\operatorname{TR}$-group conjecture in degree 9.

$$\operatorname{TR}^3_{9-\lambda_4}(\mathbb{Z};2)\cong\mathbb{Z}/2\oplus\mathbb{Z}/16.$$

The computation is based on the conjectured behavior of the $C_4$-Tate spectral sequence in total degrees $9$ and $10$; the source explains that the longer differentials have not been completely ruled out.