inequality – Sum of the power tower that tends to $ 1 $

Inspired by an inequality of Vasile Cirtoaje I have this:

Leave $ 0 <x <1 $ and define the function:
$$ f (y) = x ^ {1-x + y} quad operator name {y} quad g (y) = (1-x) ^ {x + y} $$
Y $$ h (x) = x ^ {2 (1-x)} $$
Now denotes by:
$ f ^ n (y) = f (f (f (f cdots (y) cdots)) $ Y $ g ^ n (y) = g (g (g (g cdots (y) cdots)) $

Where we compose $ n $ times the functions $ f (y), g (y) $ with himself.$ n $ It is an even number.

Then :
$$ f ^ n (h (x)) + g ^ n (h (1-x)) leq1 $$

Example:

by $ n = $ 2 we have :

$$ f (f (f (h (x))) = x ^ {1-x + x ^ {1-x + x ^ {1-x + h (x)}}} $$

Y

$$ g (g (g (h (1-x)))) = (1-x) ^ {x + (1-x) ^ {x + (1-x) ^ {x + h (1-x )}}} $$

I try to use the Lambert function but it doesn't solve my energy tower.

The main comment is when $ n $ tends to infinity, the terms of inequality tend to $ x $ Y $ 1-x $

If you have a good idea, thank you in advance for sharing it.