The Beatty-sequence logarithm bound by $1$

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

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

Numerical data in the source suggest this stronger bound than the preceding conjecture. The source does not provide a proof or resolution.