How to show $2+\left(\frac{1}{2}\left(\sqrt{\frac{e^{e^{-1}}}{e^{-e^{-1}}}}+\sqrt{\frac{e^{-e^{-1}}}{e^{e^{-1}}}}\right)\right)^{2}<\pi$?

Question

Problem :

$$2+\left(\frac{1}{2}\left(\sqrt{\frac{e^{e^{-1}}}{e^{-e^{-1}}}}+\sqrt{\frac{e^{-e^{-1}}}{e^{e^{-1}}}}\right)\right)^{2}<\pi$$

Some related work :

You can find some material here Showing $\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$ without a calculator .

I also tried to play with :

$$\sqrt{x}+\frac{1}{\sqrt{x}}\leq x+\frac{1}{x}$$

But it's really not satifactory .

As remark there is a link with the function $f(x)=x^x$ and the extrema .

In fact I found it in trying to show this problem Prove that $\int_0^\infty\frac1{x^x}\, dx<2$ using Reverse Cauchy Schwarz for integrals .

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Edit :

We an interesting inequality here How can you show by hand that $ e^{-1/e} < \ln(2) $?

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How to show it without a calculator ?

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Thanks and Im quite excited by your answer .