Unconventional Inequality $ \frac{x^x}{|x-y|}+\frac{y^y}{|y-z|}+\frac{z^z}{|z-x|} > \frac72$

$x,y,z >0$, and $x \neq y \neq z$, prove $$ \frac{x^x}{|x-y|}+\frac{y^y}{|y-z|}+\frac{z^z}{|z-x|} > \frac72$$

I never see this kind of inequality in any textbook yet. No idea whatsoever to proceed. This inequality has both absolute and tetration functions. I think only weight AM-GM may work.

DISCLAIMER: this is not a proof, but it was impossible to write this in a comment format. Hope it has some use...

Minimum is reached for: $$(x,y,z) = (0.043704, 0.74407, 1.5934).$$

The minimum value is $3.5458 > \frac{7}{2}$.

Running the following script in Octave using command $source("file")$ will give you the above (note that in the script $x=x(1),y-x=x(2),z-y=x(3)$):

function r = g(x)
 r = x(1)^x(1)/x(2) + (x(1)+x(2))^(x(1)+x(2))/x(3) +(x(1)+x(2)+x(3))^(x(1)+x(2)+x(3))/(x(2)+x(3));
endfunction


function r = h(x)
 r = [x(1);x(2);x(3)];
endfunction

lb = [1e-5; 1e-5; 1e-5]; ub = [10;10;10]; x0 = [1; 1; 1];

[x, obj, info, iter, nf, lambda]= sqp (x0, @g, [], @h, lb, ub, 200, 1e-15) $