Lipschitz nodal-set conjecture for Neumann eigenfunctions on symmetric convex bodies

Let $\Omega\subset\mathbb{R}^n$ be bounded, open, convex, and symmetric, and let $u$ be a Neumann eigenfunction with the least non-zero eigenvalue. Lipschitz nodal-set conjecture. There are suitable coordinates and a Lipschitz function $\varphi:\mathbb{R}^{n-1}\to\mathbb{R}$ such that

$$\{x\in\Omega:u(x)=0\}=\{(x',x_n)\in\Omega:x_n=\varphi(x')\},\qquad |\nabla\varphi|\le c_n.$$

This quantitative nodal-set statement is proposed as an ingredient for a continuity argument proving the hot spots conjecture; the source presents it as open and notes that quantitative control is needed near the extrema.