The Beatty-sequence logarithm bound by $\pi^2/6$

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound.

$$\left| \log\frac{a}{b}-\sum_{n=1}^{\infty}\frac{a \{a^{-1} (n+1)\}-b\{b^{-1}(n+1)\}}{n(n+1)}\right|<\frac{\pi^2}{6}.$$

This conjecture is motivated by estimating the corresponding series through the quantities $\frac{\pi^2}{6a}$ and $\frac{\pi^2}{6b}$. Its status is not resolved in the supplied source.