  In this paper, we establish the Anderson localization, strong dynamical
localization and the $(\frac 12-)$-H\"older continuity of the integrated
density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP)
Schr\"odinger operators with asymmetric $C^2$-cosine type potentials. We extend
both the iteration scheme of \cite{CSZ23a} and the interlacing method of
\cite{FV21} to handle asymmetric Rellich functions with collapsed gaps.
