The common horizontal tangent multiplicity conjecture for resultants

Let $f_1,f_2$ define two affine plane curves, and let $g$ generate their elimination ideal in the $y$-coordinate, while $R$ denotes their resultant with respect to $x$. Assume that no two affine roots of the system given by $f_1$ and $f_2$ have the same $y$-coordinate. Common horizontal tangent multiplicity conjecture. If the two curves defined by $f_1$ and $f_2$ admit a common tangent at an intersection point $P$ that is parallel to the $x$-axis, then the multiplicity of the factor corresponding to the projection of $P$ in $g$ is strictly smaller than the multiplicity of the factor corresponding to $P$ in $R$. The conjecture describes a source of multiplicity differences between the resultant and a generator of the elimination ideal; its status is presented in the source as an open problem for future work.