Square-full values of quadratic polynomials

A $\mathit{\text{square-full}}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n⩽N$ is denoted by $S_f^{\phantom{(}\mathrm{■}}(N)$. We have known that for a relatively prime pair $(a,b)\in \mathbb{N}\times \mathbb{N}\cup \{0\}$ with a linear polynomial $f(x)=ax+b$, its counting function is ${\asymp }_{a,b}{N}^{\frac{1}{2}}$. Fix $\epsilon >0$, for an admissible quadratic polynomial $f(x)$, we prove that $$S_f^{\phantom{(}\mathrm{■}}(N){\ll }_{\epsilon ,f}{N}^{\varpi +\epsilon }$$ for some absolute constant $\varpi <1\mathrm{/}2$. Under the assumption on the $abc$ conjecture, we expect the upper bound to be $O_{\epsilon ,f}(N^{\epsilon })$.