Larson–Vogt conjecture on signed real bitangent counts of plane quartics

Let $V(Q)$ be a real quartic such that for each bitangent $L$, $L\cap V(Q)\cap L_\infty=\emptyset$, where $L_\infty$ denotes the line at infinity. For a real bitangent $L$, let $\mathrm{Qtype}_{L_{\infty}}$ denote its real quadratic type, which is either $\langle 1\rangle$ or $\langle -1\rangle$. Larson–Vogt's conjecture. The number of real bitangents with $\mathrm{Qtype}_{L_{\infty}}$ equal to $\langle 1\rangle$ minus the number of real bitangents with $\mathrm{Qtype}_{L_{\infty}}$ equal to $\langle -1\rangle$ is in

$$\{0,2,4,6,8\}.$$

Larson and Vogt formulated this conjecture based on a randomized search. It refines the signed count of real bitangents according to their quadratic type; the source reports the corresponding difference as $4$ in the stated setting, while the conjectured range describes the values observed in the broader computational investigation. Its resolution is not given here.