Uniqueness conjecture for strong-KPP steady states on bounded Lipschitz domains

Let $f$ be a strong-KPP reaction, let $\Omega$ be a bounded Lipschitz domain, and let $\varrho=0$. Consider the steady-state problem

$$\text{equation (main)}$$

Uniqueness conjecture. The problem admits at most one solution.

The preceding uniqueness proof uses the Hopf lemma and therefore the interior ball property. Uniqueness in less regular domains is stated to be open; the conjecture proposes that Lipschitz regularity suffices.