Conjecture on the signed count of real bitangents to a plane quartic

Let $Q$ be a smooth plane quartic defined over $\mathbb{R}$, and let $L_\infty \subset \mathbb{P}^2_{\mathbb{R}}$ be a line defined over $\mathbb{R}$ such that $L \cap Q \cap L_\infty = \varnothing$ for every bitangent $L$. For a real bitangent $L$, let $\operatorname{Qtype}_{L_\infty}(L)$ denote its quadratic type relative to $L_\infty$. Signed-count conjecture. The difference between the numbers of real bitangents of the two indicated quadratic types satisfies

$$\#\left(\{\text{real bitangents with }\operatorname{Qtype}_{L_\infty}(L)=\langle 1\rangle\}\right)-\#\left(\{\text{real bitangents with }\operatorname{Qtype}_{L_\infty}(L)=\langle -1\rangle\}\right)\in\{0,2,4,6,8\}.$$

This conjecture is based on a randomized search of more than $10000$ quartics and predicts the range of signed counts obtainable by varying the line at infinity for a fixed smooth real plane quartic.