Today I have this to propose :
Let $a,b,c,d$ be real positive numbers such that $a+b+c+d=4$ and $a\geq 3>1\geq b\ge c \geq d$ and $0\leq\varepsilon\leq d$ then we have : $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq (a+2\varepsilon)^{(a+2\varepsilon)(b-\varepsilon)}+(b-\varepsilon)^{(b-\varepsilon)c}+c^{c(d-\varepsilon)}+(d-\varepsilon)^{(d-\varepsilon)(a+2\varepsilon)} $$
My try :
I want to prove that we have $f(\varepsilon)$ convex with :
$$f(\varepsilon)=(a+2\varepsilon)^{(a+2\varepsilon)(b-\varepsilon)}+(b-\varepsilon)^{(b-\varepsilon)c}+c^{c(d-\varepsilon)}+(d-\varepsilon)^{(d-\varepsilon)(a+2\varepsilon)} $$
And work with this but I can't go further .
Any hints would be appreciable
Thanks.
Edit : it's a conjecture but with the condition of the minimum I mean , $3\leq a\leq 3.3$ and $0\leq b\leq 0.5$ and $b\geq c \geq d $ we have the following refinement : $$a^{ab}+b^{bc}+c^{cd}+d^{da}> \sum_{cyc}e^{\frac{-a^2}{2.55^2}}> \pi$$
@Simplifire Thanks for the invitation I will speak with you the next week if you want :) . And yes we are working on the same things . Have a good day .