Conjecture on the range of real bitangents of smooth real plane sextics

Let $C$ be a smooth sextic curve in the real projective plane $\mathbb{P}^2_\mathbb{R}$. A bitangent of $C$ is a line tangent to $C$ at two points, and a bitangent is real when it is defined over $\mathbb{R}$. Real-bitangent range conjecture. The number of real bitangents of $C$ ranges from $12$ to $306$. The lower bound is attained by curves of types empty, ${\rm 1}$, ${\rm 2}$, ${\rm (11)}$, and ${\rm (hyp)}$, while the upper bound is attained by certain $11$-oval curves of Gudkov type $(51)5$. This conjecture concerns the possible extremal numbers of real bitangents among the rigid isotopy types of smooth real sextics; the cited experiments identify examples attaining both bounds, but no resolution beyond the experimental evidence is supplied here.