Convergence to the minimal-speed front along level sets

Let $u$ be the solution of the homogeneous reaction-diffusion equation with initial datum as in the paper, let $X_\lambda(t,x')$ be its $\lambda$-level set, and let $\varphi$ be the traveling-front profile with minimal speed $c^*$. Assume the hypotheses of Theorems

$$and$$

. Convergence-along-level-sets conjecture. For every $\lambda\in(0,1)$, every sequence $t_n\to+\infty$, and every bounded sequence $x'_n\in\mathbb{R}^{N-1}$,

$$u(t_n+t,x'_n+x',X_\lambda(t_n,x'_n)+x_N)\longrightarrow\varphi\bigl(x_N-c^*t+\varphi^{-1}(\lambda)\bigr)$$

as $n\to+\infty$ in $C^{1;2}_{\mathrm{loc}}(\mathbb{R}_t\times\mathbb{R}^{N-1}_{x'})$ and uniformly in $x_N\in\mathbb{R}$. If additionally $\lim_{|x'|\to+\infty}\nabla\gamma(x')=0$, the same convergence should hold for arbitrary sequences $x'_n\in\mathbb{R}^{N-1}$. The claim describes local convergence to the one-dimensional minimal-speed front, extending the asymptotic planar behavior established by the surrounding theorems.