A conjectured upper bound for $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n$ and $x\geq 1$

Question

Hi I have (related https://mathoverflow.net/questions/337457/prove-that-left-fracxn1xn-11-rightn-left-fracx12-rightn ):

Let $x\geq 1$ a real number and $n\geq 2$ a natural number then we have : $$\Big(\frac{x^{n+1}+1}{x+1}\Big)\Bigg(\frac{\Big(\frac{x+1}{2}\Big)^n}{\Big(\frac{x^2+1}{x+1}\Big)^n}+\frac{(x^n+1)^2}{(x^{n-1}+1)(x^{n+1}+1)}\Bigg)\geq \left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n$$

It's a conjecture so I have tested for $n\leq 20$ . Maybe it works also for $n$ a positive real number . If it's true I think we can make something similar to the Peter Mueller's answer. For the case $n=3$ here we have the derivative . Clearly the numerator (of the derivative) is positive for $x\geq 1$.But in some way it's not so easy with the general case .

So how to solve it ?

Any helps is greatly appreciated.

Thanks in advance for all your contribution .