Purely imaginary roots under reverse-reaction removal

Let $G$ be a reaction network containing the motif

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

where $Y$ appears in no other complex. Let $p_{(h,\ell)}$ and $p'_{(h,\ell')}$ be the reduced characteristic polynomials of $G$ and the network obtained by removing $Y\to y$, with $h\in\mathbb{R}^n_{>0}$, $\ell\in\mathbb{R}^m_{>0}$, and $\ell'\in\mathbb{R}^{m-1}_{>0}$. Purely imaginary-root equivalence conjecture. The polynomial $p_{(h,\ell)}$ has purely imaginary roots for some $(h,\ell)\in\mathbb{R}^n_{>0}\times\mathbb{R}^m_{>0}$ if and only if $p'_{(h,\ell')}$ has purely imaginary roots for some $(h,\ell')\in\mathbb{R}^n_{>0}\times\mathbb{R}^{m-1}_{>0}$. Such roots are relevant because they are necessary for the Hopf-bifurcation criterion, while the conjecture itself remains unproved.