Kernelization hardness for crossing number with prescribed rotation systems

Let $G$ be a graph with a given rotation system, and let $k\geq 1$ be an integer. A drawing of $G$ respecting the prescribed rotation system is one in which the clockwise order of the edges incident with every vertex agrees with the given rotation. The crossing number problem asks whether such a drawing has at most $k$ crossings. Kernelization-hardness conjecture. The problem of deciding whether $G$ has a drawing respecting the prescribed rotation system with at most $k$ crossings, parameterized by $k$, does not admit a polynomial kernel unless $\operatorname{NP}\subseteq\operatorname{coNP/poly}$. Consequently, the crossing number problem restricted to cubic graphs, and the analogous minor crossing number problem, do not admit a polynomial kernel with respect to $k$ unless $\operatorname{NP}\subseteq\operatorname{coNP/poly}$. The paper proves the corresponding lower bound for almost-planar graphs without prescribed rotation systems; this proposed strengthening would extend the obstruction to graphs with prescribed rotations and, consequently, to cubic and minor crossing number. Its status is left open by the source.