The paper is accepted following a long discussion post rebuttal. Reviewers believe there is value in this paper, since it provides an improved result over the currently known bounds of the Sinkhorn algorithm using a different strategy. I do however share the concern expressed by most reviewers that the experimental section is underwhelming. The paper would greatly benefit from taking more seriously this aspect into account. In the setup considered by the authors the Sinkhorn algorithm can easily scale to 10k x 10k grids using the convolution trick, or in 3D in 200 x 200 x 200. The scales presented here are too small to be convincing. We understand this is not the main ingredient of the paper, but if the paper is to be presented at Neurips a diligent experimental evaluation is a big plus for the community.