Bi-Lipschitz reduction conjecture for generalized outward cuspidal domains

Let $U$ be a bounded Lipschitz domain in ${\mathbb R}^{n-1}$ with $0\in U$, and let $\psi:(0,1]\to(0,\infty)$ be a cuspidal function. Define the generalized outward cuspidal domain

$$U_\psi^n:=\left\{(t,x)\in(0,1]\times{\mathbb R}^{n-1}:x\in\psi(t)U\right\}\cup\left\{(t,x)\in(1,2)\times{\mathbb R}^{n-1}:x\in\psi(1)U\right\}.$$

Bi-Lipschitz reduction conjecture. For every generalized outward cuspidal domain $U_\psi^n$, there exists a Lipschitz cuspidal function $\hat\psi$, a Lipschitz base domain $\hat U$, and a global bi-Lipschitz transformation $\mathcal O:{\mathbb R}^n\to{\mathbb R}^n$ such that

$$\mathcal O(U_\psi^n)=\hat U_{\hat\psi}^n.$$

Such a reduction would extend the results for outward cuspidal domains with unit-ball bases to generalized domains with arbitrary Lipschitz bases. The source describes establishing the required global bi-Lipschitz equivalence as technically challenging but expects it to be doable; no proof or resolution is given here.