The complete monotonicity conjecture for a logarithmic ratio

For $\tau\geq0$, define

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

Complete monotonicity conjecture. The function $h_1$ is completely monotonic on $(0,\infty)$. This is a stronger regularity assertion about a logarithmic ratio related to the gamma-function monotonicity problems in the paper. The supplied status evidence marks the associated conjectural result as resolved.