The problem is indeed misquoted. Here is the correct statement:

If a polynomial $P(x)$ with rational coefficients 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 $yz\neq 0$.

(copied verbatim from the text)

The problem is indeed misquoted. Here is the correct statement:

If a polynomial $P(x)$ with rational coefficients 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 $yz\neq 0$.

The paper can be found here.