Threshold conjecture for the EpiSLFV reproduction number

Let $\gamma>0$, and let $\nu$ be a $\sigma$-finite measure on $(0,\infty)\times(0,1]$ satisfying the source condition. For the $(\gamma,\nu)$-EpiSLFV process, define the reproduction number by

$$\mathrm{R}_{0}(\gamma,\nu):=\frac{1}{\gamma}\int_{0}^{1}\int_{0}^{\infty}uV_{r}\,\nu(dr,du).$$

Threshold conjecture. For all such $\gamma$ and $\nu$: if $\mathrm{R}_{0}(\gamma,\nu)<1$, then the $(\gamma,\nu)$-EpiSLFV process goes extinct; if $\mathrm{R}_{0}(\gamma,\nu)>1$, then the $(\gamma,\nu)$-EpiSLFV process does not go extinct.