Erdős problem 884

For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of n in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty`$, i.e. \sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) \right)?