For entropic unbalanced optimal transport problem, it is well known that the Sinkhorn algorithm converges slowly for relatively small regularization. Possible improvements exist but still usually suffer from similar, though less severe limitation. In this talk, we present INexact Promixal Unbalanced optimal Transport (INPUT) - a simple, yet useful extension of the Inexact Promixal Optimal Transport (IPOT) to the unbalanced setting. As a byproduct, this algorithm can also naturally handle the unregularized problem. We illustrate the promising empirical acceleration of INPUT versus other existing approaches on toy examples. By contrast, despite being an immediate generalization of IPOT, the convergence of INPUT remains little understood since the technique used to study IPOT can not be adapted to the unbalanced problem. We hope that the discussion from this seminar may shed some light on the theoretical analysis.