Hopf-bifurcation invariance under removing a reverse reaction

Let $G$ be a reaction network with $n$ species containing a motif

$$y\rightleftarrows Y\to y',$$

such that the species $Y$ does not appear in any other complex of the network. Let $G'$ be obtained from $G$ by removing the backward reaction $Y\to y$. Hopf-bifurcation reduction conjecture. The network $G$ admits a Hopf bifurcation in $\mathbb{R}^n_{>0}$ if and only if $G'$ admits a Hopf bifurcation in $\mathbb{R}^n_{>0}$. This is a precise form of the proposed reduction principle; the paper provides evidence for it but does not establish it in general.