The monotonicity conjecture for the gamma logarithmic ratio

Let $\Gamma$ denote the gamma function and let $\gamma$ be the Euler–Mascheroni constant. For $\tau\geq0$, define

$$g_\tau(x)= \begin{cases} \dfrac{\ln\Gamma(x)}{\ln(x^2+\tau)-\ln(x+\tau)},&x\ne1,\\ -(1+\tau)\gamma,&x=1. \end{cases}$$

The monotonicity conjecture. For every $\tau\geq0$, the function $g_\tau$ is strictly increasing on $(0,\infty)$. This conjecture is one of the monotonicity problems arising from refinements of gamma-function inequalities; it was confirmed by the cited theorem.