  We investigate the stability of the equilibrium-induced optimal value in
one-dimensional diffusion setting for a time-inconsistent stopping problem
under non-exponential discounting. We show that the optimal value is
semi-continuous with respect to the drift, volatility, and reward function. An
example is provided showing that the exact continuity may fail. With equilibria
extended to $\varepsilon$-equilibria, we establish the relaxed continuity of
the optimal value.
