Resonance-pole instability conjecture for inhomogeneous coupling

Consider the modified system

$$\begin{cases} \partial_t u=d\partial_{xx}u+s\partial_xu+f(u)+\beta\sigma(x)v,\\ \partial_t v=-(\partial_{xx}+1)^2v+s\partial_xv+\mu v, \end{cases} \qquad t>0,\quad x\in\mathbb{R},$$

where the coupling term $\beta v$ has been replaced by $\beta\sigma(x)v$, and suppose that the Fourier transform of $\sigma(x)$ has full support. Resonance-pole instability conjecture. For parameters in $\mathcal{R}_{\mathrm{abs}}$, the traveling wave $(Q_*(x),0)$ is pointwise unstable due to resonance poles accumulating on the unstable absolute spectrum $\Sigma_{\mathrm{abs}}$. This conjecture proposes a mechanism by which spatial inhomogeneity induces instability in a front that is stable in the corresponding homogeneous setting; the source presents numerical evidence and formulates the claim as a conjecture.