Median monotonicity and bounds conjecture for the variance-gamma distribution

Let $V_{r,\theta,\sigma}\sim\operatorname{VG}(r,\theta,\sigma,0)$, with $r>0$ and $\theta>0$. Let $G\sim\Gamma(r/2,(2\theta)^{-1})$.

Variance-gamma median conjecture. The function $\sigma\mapsto\operatorname{Med}(V_{r,\theta,\sigma})$ is strictly decreasing for $\sigma\in(0,\infty)$, and consequently

$$\operatorname{Med}(V_{r,\theta,\sigma})<\operatorname{Med}(G).$$

Moreover,

$$\lim_{\sigma\to\infty}\operatorname{Med}(V_{r,\theta,\sigma})=(r-1)\theta\quad(r>1),$$

and the limit is $0$ for $r\leq1$. Thus, for $r>0$,

$$(r-1)\theta<\operatorname{Med}(V_{r,\theta,\sigma})<r\theta e^{-2/(3r)}<\left(r-\frac23+\frac{2}{9r}\right)\theta,$$

and, for $r\geq2$,

$$\operatorname{Med}(V_{r,\theta,\sigma})\leq(r+2\log 2-2)\theta.$$

The source presents these as conjectured monotonicity, limiting behavior, and bounds for the variance-gamma median; no resolution is supplied.