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The main challenge of monocular 3D object detection is the accurate localization of 3D center. Motivated by a new and strong observation that this challenge can be remedied by a 3D-space local-grid search scheme in an ideal case, we propose a stage-wise approach, which combines the information flow from 2D-to-3D (3D bounding box proposal generation with a single 2D image) and 3D-to-2D (proposal verification by denoising with 3D-to-2D contexts) in a topdown manner. Specifically, we first obtain initial proposals from off-the-shelf backbone monocular 3D detectors. Then, we generate a 3D anchor space by local-grid sampling from the initial proposals. Finally, we perform 3D bounding box denoising at the 3D-to-2D proposal verification stage. To effectively learn discriminative features for denoising highly overlapped proposals, this paper presents a method of using the Perceiver I/O model [20] to fuse the 3D-to-2D geometric information and the 2D appearance information. With the encoded latent representation of a proposal, the verification head is implemented with a self-attention module. Our method, named as MonoXiver, is generic and can be easily adapted to any backbone monocular 3D detectors. Experimental results on the well-established KITTI dataset and the challenging large-scale Waymo dataset show that MonoXiver consistently achieves improvement with limited computation overhead.