Conjecture on the middle-degree Bockstein and injectivity of the second Bockstein

Let $N$ be an integer, let $B_1$ and $B_2$ be the differentials in the spectral sequence under consideration, and let $HH_i^\sigma(\mathcal{A})$ denote the twisted Hochschild homology groups. For $0\leq i\leq N$, write $\omega_1(N,i)$ and $\omega_2(N,i)$ for the indicated generators, and let $f:\operatorname{ker}(B_0)\to HH_3^\sigma(\mathcal{A})$ be the map appearing in the spectral sequence.

Bockstein conjecture. For $N$ even,

$$[B_1(\omega_1(N,\tfrac{1}{2}N))]\neq 0\in HH_2^\sigma(\mathcal{A}),$$

and this class is proportional to $[\omega_2(N,\tfrac{1}{2}N)]$. For all $N$, the map

$$B_2:HH_2^\sigma(\mathcal{A})/\operatorname{im}(B_1)\longrightarrow HH_3^\sigma(\mathcal{A})$$

is injective. Consequently, for $N$ odd, the map

$$f:\operatorname{ker}(B_0)\longrightarrow HH_3^\sigma(\mathcal{A})$$

is zero.

These expected properties would determine the relevant differentials in the spectral sequence and hence clarify the computation of the twisted cyclic homology groups. The source explicitly states that no proof is currently available; the status of both assertions is therefore open.