It is not difficult to prove that if $x,y\in\mathbb{R}^+$ the inequality $$ \frac{x+y}{2}+\frac{2}{\frac{1}{x}+\frac{1}{y}}\geq \color{purple}{2}\cdot\sqrt{xy} $$ holds, and the constant $\color{purple}{2}$ is optimal.

In a recent question I proved, with a quite involved technique, that if $x,y,z\in\mathbb{R}^+$ then $$ \frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}\geq \color{purple}{\frac{5}{2\sqrt[3]{2}}}\cdot\sqrt[3]{xyz} $$ holds, and the constant $\color{purple}{\frac{5}{2\sqrt[3]{2}}}$ (that is a bit less than $2$) is optimal. Then I was wondering:

Given $x_1,x_2,\ldots,x_n\in\mathbb{R}^+$, what is the optimal constant $C_n$ such that: $$\text{AM}(x_1,\ldots,x_n)+\text{HM}(x_1,\ldots,x_n)\geq \color{purple}{C_n}\cdot \text{GM}(x_1,\ldots,x_n)$$

I do not think my approach with $3$ variables has a simple generalization (also because in $\mathbb{R}_+^3$ the stationary points are non-trivial), but maybe something is well-known about the improvements of the AM-GM inequality, or there is a cunning approach by some sort of induction on $n$.