Transience conjecture for the two-environment reaction-network example

Consider the two-environment mixed process described in Example: environment $1$ uses the mass-action reaction network $\mathcal R_1$, environment $2$ uses $\mathcal R_2$, and the process switches between the environments at rate $\kappa$ in each direction. Let $\alpha>0$ be the parameter appearing in those reaction networks.

Example transience conjecture. There exists $\alpha>0$ such that the mixed process from Example is transient for all sufficiently small values of $\kappa$.

This is presented as a potentially more tractable special case of the common-support conjecture. The source states that the authors cannot currently prove or disprove the general conjecture, and does not resolve this special case, so it remains open.