Lower bound for cyclic sums of Diananda type

Let $C=\inf (k/n)\sum_{i=1}^n x_i(x_{i+1}+\dots+x_{i+k})^{-1}$, where the infimum is taken over all pairs of integers $n\geq k\geq 1$ and all positive $x_1,\dots,x_{n+k}$ subject to cyclicity assumption $x_{n+i}=x_i$, $i=1,\dots,k$. We prove that $\ln 2\leq C< 0.9305$. In the definition of the constant $C$ the operation $\inf_k\inf_n\inf_{\mathbf{x}}$ can be replaced by $\lim_{k\to\infty}\lim_{n\to\infty}\inf_{\mathbf{x}}$.