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According to Lemma 3.1 of Lagarias's paper in which he introduces his criterion for RH, one has $$ e^{H_n} \log H_n \geq e^{\gamma} n \log \log n$$ for any $n \geq 3$. This means that any $n$ which violates your inequality would also yield a counterexample to Robin's inequality $$ \sigma (n) < e^\gamma n \log \log n.$$ Since it is known that the truth of RH implies that $5040$ is the largest counterexample, no $n$ for which $\sigma(n) >e^{H_n}\log H_n$ greater than $5040$ should be found. A computer check then confirms that $n=60$ is the largest counterexample. The interest in Lagarias's criterion lies in its requirement being more restrictive than Robin's (and, of course, than the inequality in this question).