Verifying a proof that $\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}≥x^2+y^2+z^2$ when $xyz=1$ and x,y,z are positive real number

Question

if $xyz=1$ and $x,y,z$ are positive real number

prove

$$ \frac{x^2 }{y^3}+\frac{y^2 }{z^3}+\frac{z^2 }{x^3}≥x^2+y^2+z^2$$

This is presumbly a middle school question, schoolmates put it on group chat, and we-engineering master degree students- still don't have an answer. I've spend couple of hours, can't solve it. I've tried method of Lagrange multipliers, but cat't get clear progress after setting derivatives to 0, but it may be simply me not grasping this method. Also tried AM-GM inequality, but it seems not applicable because both side are the form of AM.

I feel it could be solved by amplification and minification, but can't think of one by myself.

First time here, please forgive my expression problems if not meeting standards of this community.

Answer 1

$$\sum_{cyc}\frac{x^2}{y^3}=\frac{1}{19}\sum_{cyc}\left(\frac{11x^2}{y^3}+\frac{7y^2}{z^3}+\frac{z^2}{x^3}\right).$$ Can you end it now?