A higher-dimensional Schrödinger maximal estimate along families of bi-Lipschitz curves

Let $Θ\subset\mathbb{R}^n$, $n\geq 1$ be compact with upper Minkowski dimension $β$, and let $γ(x,t,θ)$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $$\left\|\sup_{(τ,θ)\in[0,1)\timesΘ}\left|e^{iτΔ}f(γ(\,\cdot\,,τ,θ))\