Assume $u$ is a smooth solution for $$ \Delta u + f(u)=0\qquad \hbox{in}\quad \Omega $$ and $\Omega$ is a smooth convex domain in $\mathbb{R}^n$.

Is there a conjecture which are the weakest conditions on f under which the solution has convex superlevels?

Background to the question: In the 1980s, there was work by Acker, Caffarelli, Friedman, Spruck and others on this topic. However, the conditions at f are very different in each case. In the meantime there is also a counterexample by Hamel, Nadirashvili and Sire for the case f=-1.