Strong ironing property for rolling-ball optimizers

Let $f,g:\mathbb{R}^d\to\mathbb{R}$ be two continuous functions, and let $\Gamma,\Gamma^\prime$ be their respective graphs. Write $\operatorname{d_{\mathrm{H}}}(\Gamma^\prime,\Gamma)$ for the Hausdorff distance between the graphs, and let $\operatorname{S}(\Gamma,\rho)$ denote the rolling-ball optimizer associated with $\Gamma$ at scale $\rho$. Two spaces are $\varepsilon$-almost isometric if there is a map between them whose distance distortion is at most $\varepsilon$. Strong ironing property. If

$$\operatorname{d_{\mathrm{H}}}(\Gamma^\prime,\Gamma)<+\infty,$$

then $\operatorname{S}(\Gamma,\rho)$ and $\operatorname{S}(\Gamma^\prime,\rho)$ are $\varepsilon_\rho$-almost isometric, with $\varepsilon_\rho\to0$ as $\rho\to+\infty$. The claim would extend the observed ironing phenomenon from affine and other controlled perturbations to arbitrary continuous functions; the supplied context states that it remains unproved and is deferred to future work.