Adiabatic vanishing conjecture for the transient current

Let $H$ be the coupled Hamiltonian, let $P^{d}(H)$ be the projection onto the subspace generated by the discrete eigenfunctions of $H$, and let ${\varrho}_{\alpha}(t)$ denote the time-dependent state for coupling parameter $\alpha>0$. Let $\chi$ be any smoothed out characteristic function of one of the reservoirs. Adiabatic vanishing conjecture.

$$\lim_{\alpha\searrow 0}\limsup_{t\to \infty}\left \vert {\rm Tr} \{{\varrho}_{\alpha} (t)P^{d}(H)[H,\chi]\}\right \vert =0.$$

This conjecture asserts that the transient-current contribution associated with the discrete spectrum vanishes in the adiabatic limit. It is motivated by the expectation that adiabatic switching removes the oscillations caused by interference between different eigenfunctions; the source presents the claim as an open problem.