Berestycki–Caffarelli–Nirenberg conjecture on symmetry in the half-space

Let $n\geq 1$, let $f\in C^1([0,\infty))$, and consider positive bounded classical solutions of

$$-\Delta u=f(u)\quad\text{in }\mathbb{R}^n_+,\qquad u=0\quad\text{on }\partial\mathbb{R}^n_+.$$

Berestycki–Caffarelli–Nirenberg conjecture. (i) If there is a positive bounded solution $u$ of this problem, then $u$ depends only on $x_n$; consequently, $u_{x_n}>0$ and $f(\sup u)=0$. (ii) In particular, if $f>0$ on $(0,\infty)$, then the problem has no positive bounded solution. This is a symmetry and one-dimensionality conjecture for elliptic equations in a half-space. The supplied text attributes it to Berestycki, Caffarelli and Nirenberg, but gives no resolution status.