Diophantine equation involving factorials

Question

$$x!+y=y^3$$ $$y=\sqrt[3]{x!+\sqrt[3]{x!+\sqrt[3]{x!+\cdots}}}$$

The only integer solutions to these identities that I have found are:

$$3!+2=2^3$$ $$4!+3=3^3$$ $$5!+5=5^3$$ $$6!+9=9^3$$

I conjecture these are all the solutions. Is that true

Answer 1

Assuming the ABC conjecture the following paper shows that there should only be finitely many solutions.

https://web.math.pmf.unizg.hr/glasnik/37.2/37(2)-04.pdf

I would probably bet that the solutions you have found are all of the solutions.