Let $|g|=\min(g,n-g)=$Lee-norm on $\mathbb{Z}/(n)=$ word length on cyclic group $C_n$ with respect to generating set $S=\{\pm 1\}$. Let $H^{L}(C_n) := \sum_{g \in C_n} \frac{1}{|g|+1}$, which as one computes is equal to $E_n$ if $n\equiv 0 \mod(2)$ and it is equal to $O_n$ if $n\equiv 1 \mod(2)$, where: $$E_n = 2 \cdot H_{n/2}-1+2/(n+2)$$ and $$O_n = 2 \cdot H_{(n+1)/2} - 1$$
In "Equivalents of the Riemann Hypothesis, 1" it is mentioned (page 196, 7.94 here), that uncoditionally
$$(1): \sigma(n) \le H_n + 2\exp(H_n)\log(H_n), n \ge 1$$ is proved, where $\sigma(n)$ is the sum of divisors and $H_n$ is the $n$-th harmonic number. For all $ n\ge 7$ it seems (numerically) to be the case, that:
$$(2): H_n+2\exp(H_n)\log(H_n) \le L(H^L(C_n))$$
where $L(x) = x+\exp(x)\log(x)$ is the Lagarias function. For $n=1,\cdots,6$ one computes that $$(3): \sigma(n) \le L(H^L(C_n))$$
From (1)-(3) it would follow that:
$$\sigma(n) \le L(H^L(C_n)), n \ge 1$$
and this would be an indication that maybe the conjecture made in the following question is not so wrong, at least for the cyclic group:
A group theoretic interpretation of Lagarias inequality
So my question is, if one can prove uncoditionally, that the inequality $(2)$ is true?
Edit: Related question for the dihedral group: https://math.stackexchange.com/questions/3224207/two-questions-about-the-dihedral-group