THH of the image of j

In a joint work with Ishan Levy, we computed the mod $(p,v_1)$ homotopy groups THH of the $(-1)$-connective covers of the $K(1)$-local sphere and its Galois extensions, i.e. the fixed point of the connective Adams summand $\ell$ by various Adams operations. This can be used to compute TCs of these rings which are counterexamples to the telescope conjecture at height $2$. In this talk, I will talk about an ongoing project with Ishan Levy where we compute the $v_1$ Bocksteins of these THHs. The proof involves the Dehn twist argument of Burklund-Hahn-Levy-Schlank and the analysis of the trivialities of various Adams operation on $\ell/(p,v_1^k)$ for varying $k$.