Threshold conjecture for propagation and blocking in funnel-shaped domains

Let $(R,\b1\alpha)\a0\in(0,+\infty)\times(0,\pi/2)$ parameterize the funnel-shaped domain $\Omega_{R,\alpha}$, and assume the functions $h$ defining these domains depend continuously on $(R,\alpha)$ in the $C^{2,\beta}_{\operatorname{loc}}(\mathbb{R})$ sense, where $0<\beta<1$. Let $u$ be the solution of the reaction-diffusion problem in $\Omega_{R,\alpha}$ with the stated past condition; complete propagation and blocking mean that $u$ satisfies the corresponding properties.

Propagation and blocking threshold conjecture. For every $R>0$, there is $\alpha_R\in(0,\pi/2]$ such that complete propagation holds for all $\alpha\in(0,\alpha_R)$, while blocking holds for all $\alpha\in[\alpha_R,\pi/2)$ when $\alpha_R<\pi/2$. For every $\alpha\in[0,\pi/2)$, there is $\rho_\alpha\in[0,+\infty)$ such that complete propagation holds for all $R>\rho_\alpha$, while blocking holds for all $R\in(0,\rho_\alpha]$ when $\rho_\alpha>0$.

This conjecture asserts sharp threshold behavior and monotonicity of propagation with respect to the funnel parameters. Together with the preceding openness and closedness results, it would give a detailed description of the propagation and blocking regions in parameter space.