Compatibility of logarithmic and ordinary Hochschild functoriality

Let $(X,D)$ and $(X',D')$ be log smooth pairs, and let $E$ be a strong Fourier–Mukai kernel on $X\times^{\log}X'$. Let $\pi:X\times^{\log}X'\to X\times X'$ be the projection and regard the proper pushforward $\pi_*E$ as a Fourier–Mukai kernel. The kernel $E$ induces a map $\phi^{HH}_E$ on logarithmic Hochschild homology, while $\pi_*E$ induces a map $\phi^{HH}_{\pi_*E}$ on ordinary Hochschild homology. Compatibility conjecture. The diagram

$$\begin{CD} {\rm HH}_*(X') @>{\phi^{HH}_{\pi_*E}}>> {\rm HH}_*(X) \\ @VVV @VVV \\ {\rm HH}^{\log}_*(X') @>{\phi^{HH}_E}>> {\rm HH}^{\log}_*(X) \end{CD}$$

commutes, where the vertical maps are induced by the counit map $\Delta^*\Delta_*\Rightarrow i^*i_*$ from the self-intersection of the diagonal to the self-intersection of the log diagonal. This would relate ordinary and logarithmic Fourier–Mukai actions through the natural comparison maps.