Absolute-monotonicity conjecture for the second derivative of an elliptic-integral difference

Let $\mathcal{K}(r)$ denote the complete elliptic integral of the first kind and define

$$D(x)=\mathcal{K}(\sqrt{x})-\ln\left(1+\frac{4}{\sqrt{1-x}}\right).$$

A function is absolutely monotonic on an interval if all of its derivatives are nonnegative there. Absolute-monotonicity conjecture for $D^{\prime\prime}$. The second derivative $D^{\prime\prime}$ is absolutely monotonic on $(0,1)$. The paper presents this as a conjecture; the supplied text gives no resolution.