Let $1=d_1<\cdots <d_{\tau (n)}=n$ be the divisors of $n$, and for $\alpha >1$ let$$h_{\alpha }(n)=\sum _i{\left(\frac{d_{i+1}}{d_i}-1\right)}^{\alpha }.$$Is it true that$$\underset{n\to \infty }{\mathrm{liminf}}{h}_{\alpha }(n){\ll }_{\alpha }1?$$