Monotonicity conjecture for the median of a weighted sum of gamma variables

Let $X_1$ and $X_2$ be independent $\Gamma(r,1)$ random variables, and define

$$Z_\alpha=\alpha X_1+(2-\alpha)X_2,\qquad \alpha>0.$$

Gamma-sum median conjecture. The function $\alpha\mapsto\operatorname{Med}(Z_\alpha)$ is non-decreasing for $\alpha\in(0,1)$. This is one of two conjectures concerning medians of sums and differences of independent gamma random variables; its status is not resolved in the supplied text.