First coniveau notch of the Dwork family and its mirror

If $X_{\lambda }$ is a smooth member of the Dwork family over a perfect field $k$, and $Y_{\lambda }$ is its mirror variety, then the motives of $X_{\lambda }$ and $Y_{\lambda }$ are equal up to motives that are in coniveau $\ge 1$. If $k$ is a finite field, this provides a motivic explanation for Wan’s congruence between the zeta functions of $X_{\lambda }$ and $Y_{\lambda }$.