Square-full values of quadratic polynomials

Abstract: A $\textit{square-full}$ 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 $n\leqslant N$ is denoted by $S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)$. 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 $\varepsilon>0$, for an admissible quadratic polynomial $f(x)$, we prove that $$S^{{\mathstrut\hspace{0.05em}\blacksquare}}f(N)\ll{\varepsilon, f} N^{\varpi+\varepsilon}$$ for some absolute constant $\varpi<1/2$. Under the assumption on the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon,f}(N^\varepsilon)$.