Logarithmic rigidity conjecture for étale motives

Let $k$ be a field with exponential characteristic, and let $\Lambda$ be an $N$-torsion ring, where $N>0$ is coprime to the exponential characteristic of $k$. Let $\eta_\sharp$ denote the functor from the derived category of sheaves on the small logarithmic étale site to the category of effective logarithmic étale motives.

Logarithmic rigidity conjecture. The functor

$$\eta_\sharp\colon \mathbf{D}(\operatorname{Shv}(k_{l\acute{e}t},\Lambda))\rightarrow \operatorname{DM}^{\mathrm{eff}}_{\log,\acute{e}t}(k,\Lambda)$$

is an equivalence.

This is a logarithmic analogue of Suslin–Voevodsky rigidity, predicting that effective logarithmic étale motives are recovered from complexes of sheaves on the small logarithmic étale site. The source gives no resolution of this conjecture.