Lower-bound conjecture for the median of the generalized hyperbolic distribution

Let $X\sim GH(\lambda,\alpha,\beta,\delta,0)$, with $\lambda>1/2$ and $\beta>0$, and let

$$\gamma^2=\alpha^2-\beta^2.$$

Generalized hyperbolic median conjecture. The median satisfies

$$\operatorname{Med}(X)>\frac{\beta}{\gamma^2}\left[\lambda-\frac12+\sqrt{\left(\lambda-\frac12\right)^2+\delta^2\gamma^2}\right].$$

The source reports numerical experiments suggesting that this lower bound holds; no proof or resolution is supplied.