Abstract: We focus on sequential optimal experimental design (sOED) for Bayesian inverse problems, where the objective is to select experimental conditions that maximize the incremental expected information gain (iEIG) between successive posterior distributions. When the posterior is non-Gaussian, this task becomes analytically intractable and computationally expensive, especially in high-dimensional settings with costly forward models. To address these challenges, we propose a scalable framework for approximately solving the sOED problem using a sharp upper bound on the iEIG. This bound serves as a guide for sOED and is efficiently evaluated through conditional measure transport combined with likelihood-informed subspace methods. We demonstrate the effectiveness of our approach through numerical examples.