An extended comment that has a chance of being useful.

First, note that if you interchange $a_k$ and $a_{k+1}$ then only the $k$-th term on the left side changes and the right side doesn't change at all.  From this we can easily see that the worst order for a given set of values is $a_1\le a_2\le\cdots \le a_n$.

Now consider adjusting both $a_k$ and $a_{k+1}$ so that the right side remains unchanged. If $a_k\mapsto a_k+\epsilon$ then $a_{k+1}\mapsto a_{k+1} - \epsilon(a_{k+1}/a_{k})^2+O(\epsilon^2)$.  If we are sitting at the worst left side for given right side, then the effect of this adjustment must be $O(\epsilon^2)$ for all $k$.  I think that this will give the worst sequence exactly by working from the last term backwards, but I'm out of time just now.

An extension of this approach is to consider adjusting three consecutive terms so that their sum and reciprocal sum both remain the same. Then only two terms of the left side will change so the total effect on the left side can be determined to obtain an explicit condition satisfied by the worst sequence.