Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind

Let \begin{equation*} f_p(x)=\frac{\mathcal{K}(\sqrt{x})}{p-\ln\sqrt{1-x}}, \end{equation*} where $\mathcal{K}$ denotes the complete elliptic integral of the first kind, and set $g_p=1/f_p$. Tian and Yang conjectured that, at $p=\ln4$, both $-f_{\ln4}^{\prime\prime\prime}$ and $g_{\ln4}$ are absolut