We show that optimal stopping surfaces (t, y) → x∗(t, y) arising from time-inhomogeneous optimal stopping problems on two-dimensional diffusions (X, Y ) are continuous, jointly in time and space, under mild monotonicity and regularity assumptions of local nature. The talk is based on a joint paper with C. Cai and J. Palczewski (SIAM J. Control Optim. 61 (3), 2023, pp.1513–1531).