Universality conjecture for threshold regions in nonlocal bistable equations

Assume that $f_a(u)=u(1-u)(u-a)$ with $0<a<1/2$. Let $\mathcal{K}$ satisfy the stated hypotheses with normalized second moment

$$\frac{1}{2}\int_{\mathbb{R}} x^2\mathcal{K}(x)\mathrm{d}x=1.$$

Let $\mathcal{R}_j$, $j=1,\ldots,5$, be the regions defined above, and let $\ell_0^*$ and $\ell_1^*$ denote the extinction and propagation thresholds. Universality conjecture. In the five regions, respectively: (i) $0<\ell_0^*=\ell_1^*<\infty$; (ii) $0<\ell_0^*<\ell_1^*<\infty$; (iii) $\ell_0^*=0$ and $0<\ell_1^*<\infty$; (iv) $0<\ell_0^*<\infty$ and $\ell_1^*=\infty$; and (v) $\ell_0^*=0$ and $\ell_1^*=\infty$. Thus the regions $\mathcal{R}_j$ are universal among the specified class of localized kernels. The claim is motivated by numerical study of the Gaussian kernel, and proving it is described as a major open problem.