Landesman's conjecture on resultant equation counts

Let $n\geqslant 2$ and let

$$N_n(B)=\#\left\{(a,b,c)\in \mathbb{Z}^3: \begin{array}{l} |a|,|b|,|c|\leqslant B\\ \operatorname{Res}(R,ax^2+bx+c)=\pm 1\\ \text{for some $R\in \mathbb{Z}[x]$ of degree $n$} \end{array} \right\}.$$

Landesman's conjecture. There exists $m\geqslant 0$ such that

$$N_n(B)\ll B^{2}(\log B)^m.$$

The conjecture predicts that, for every fixed degree $n\geqslant 2$, the quadratic polynomials counted by the resultant equation have essentially quadratic growth, up to a power of $\log B$. The paper proves an upper bound of order $B^3/\sqrt{\log B}$ when the degree of $R$ is odd, while the conjectured bound remains open in general.