Gómez-Reñasco–López-Gómez multiplicity conjecture for indefinite boundary-value problems

Let $a(x)$ have $n+1$ intervals on which it is positive, separated by $n$ intervals on which it is negative. Consider the boundary-value problem

$$-u”=a(x)u^2+\lambda u\quad\text{in }(0,1),\qquad u(0)=u(1)=0.$$

Gómez-Reñasco–López-Gómez conjecture. There exists $\lambda_c<pi^2$ such that, for every $\lambda<\lambda_c$, the problem admits at least $2^{n+1}-1$ positive solutions.

This conjecture predicts the full combinatorial multiplicity of positive solutions for sufficiently negative parameter in a one-dimensional superlinear indefinite problem. The paper states that it remains open, although related multiplicity results are known when the parameter is zero and the negative part of the weight is sufficiently large.