Integrals,tetration, and inequalities such that $\int_{0}^{\infty}x^{-x^{\left(\frac{1}{2}x\right)^{\left(\frac{1}{3}x\right)}}}dx>2$

Question

Problem :

Prove or disprove that :

$$\int_{0}^{\infty}x^{-x^{\left(\frac{1}{2}x\right)^{\left(\frac{1}{3}x\right)}}}dx>2$$

And so on .

I have not really an attempt because my answer in the link above doesn't propose an issue .

On the other hand it seems that the differents integrals increases as the numbers of exponents increases .

Edit :

As an attempt we can try to use the integral analogue for the Carleman's inequality see https://en.wikipedia.org/wiki/Carleman%27s_inequality .

$$\int _{0}^{\infty }\exp \left\{{\frac {1}{x}}\int _{0}^{x}\ln f(t)dt\right\}dx\leq e\int _{0}^{\infty }f(x)dx$$

For any $f\geq 0$

As second attempt we are interested by :

$$\frac{1}{2}\int_{0}^{4}\left(\left(\frac{1}{x}\int_{0}^{x}\frac{1}{\left(f\left(t\right)\right)^{p}}dt\right)^{1/p}\right)dx$$

Where :

Then as $p$ increases it seems we got $2$ as limit/bound.

Edit 28/02/2022 :

We can also try straightforward method as linear approximation :

By example for the first integral inequalities it seems we have on $x\in [1,2]$ :

Where :

Edit :

We can use something like :

$$\int_{0}^{1}f\left(x\right)dx\geq \exp\left(\int_{0}^{1}\ln f\left(x\right)dx\right)$$

Where $$f(x)=x^{-x^{\left(\frac{1}{2}x\right)^{\left(\frac{1}{3}x\right)^{\left(\frac{1}{4}x\right)^{\left(\frac{1}{5}x\right)}}}}}$$

It's sufficient in this case .

Question :

How to (dis)prove it ?

Thanks for all your effort !

My attempt is too weak see here for better result https://math.stackexchange.com/questions/242123/about-a-possible-hardy-type-inequality-for-negative-exponents — Miss and Mister cassoulet char, Feb 27 '22 at 12:56