Let $H_m$ denote the $m$-th harmonic number (with the convention $H_0:=0$). Fix an integer $n$. Define for $k=0,1,\dots,n-1$ $$ d_{n,k}:={1\over{n^2}}\biggl\{\sum_{j=0}^k \bigl(H_n-H_k+H_{n-1}-H_{n-k+j-1} \bigr)^2\\ + \sum_{j=k+1}^{n-1} \bigl( H_n-H_j+H_{n-1}-H_{n-k-1} \bigr)^2\biggr\}, $$ and also $$ R_n:=\sum_{k=0}^{n-1} d_{n,k}. $$
If it exists, what is $$\lim_{n\to\infty} R_n?$$
Numerical evidence suggests that it might be around $1.28987$.