Schinzel–Tijdeman conjecture for the equation $y^2z^3=P(x)$

Let $P$ be a polynomial with rational coefficients. Schinzel–Tijdeman conjecture. If $P$ has at least three simple zeros, then the equation

$$y^2z^3=P(x)$$

has only finitely many solutions in integers $x,y,z$ with $yz\ne0$. This is presented as a second well-known conjecture for deriving conditional results about powerful numbers; its status is not resolved in the supplied text.