Threshold-location conjecture for the exponential kernel

Assume that $\mathcal{K}(x)=\mathrm{e}^{-|x|}/2$ and that $f_a(u)=u(1-u)(u-a)$ with $0<a<1/2$. Let $(a,d)\in \mathcal{R}_2\cup\mathcal{R}_3$ such that $0<\ell_1^*<\infty$ is well defined from Proposition~. Let $x_0^*>0$ be the right point of discontinuity of the discontinuous ground state solution associated to $v_0=a$. Threshold-location conjecture. Then we have $\ell_1^*=x_0^*$. This conjecture is supported by numerical agreement between direct simulations and the formula for $x_0^*$; no resolution is supplied here.