Median monotonicity and bounds conjecture for the McKay Type I distribution

Let $\phi=bc/(c^2-1)$, and let $Z_{m,c,\phi}$ be a McKay Type I random variable with parameters $m$, $b$, and $c$. Fix $m>-1/2$ and $\phi>0$, and let

$$G_1\sim\Gamma(m+1/2,(2\phi)^{-1}),\qquad G_2\sim\Gamma(2m+1,1/\phi).$$

McKay Type I median conjecture. The function $\phi\mapsto\operatorname{Med}(Z_{m,c,\phi})$ is strictly increasing for $c\in(1,\infty)$. Consequently,

$$\operatorname{Med}(G_1)<\operatorname{Med}(Z_{m,c,\phi})<\operatorname{Med}(G_2).$$

For $m>-1/2$,

$$(2m+1-\log2)\phi<(2m+1)\phi e^{-\log2/(2m+1)}<\operatorname{Med}(Z_{m,c,\phi})< (2m+1)\phi e^{-1/(3(2m+1))}<\left(2m+\frac23+\frac{1}{18(2m+1)}\right)\phi,$$

and, for $m\geq1/2$,

$$(2m+1/3)\phi<\operatorname{Med}(Z_{m,c,\phi})\leq(2m+\log2)\phi.$$

The source gives these as conjectured monotonicity and bounds, with no resolution supplied.