Existence of planar pulsating traveling waves with whole-space periodicity

Let $e\in\mathbb{S}^{n-1}$ and let $c_e^\star$ denote the critical speed in direction $-e$. A planar pulsating traveling wave solution has speed $c$, propagates in direction $-e$, and is periodic in time and in the whole spatial domain. Existence conjecture. For every $e\in\mathbb{S}^{n-1}$ and every $c\geq c_e^\star$, there exists a planar pulsating traveling wave solution in direction $-e$ with speed $c$ with time and whole-space periodicity. The preceding results establish this existence for rational directions and speeds, including the critical speed when it is rational; the conjecture asks whether the rationality restrictions can be removed.