Torus-type conjecture for reduced sextics with large essential rank

Let $C$ be a reduced plane sextic, and let $\rho(5)$ denote the rank invariant associated with the essential singularities of $C$ at level $5$.

Torus-type conjecture. If

$$\rho(5)\geq 7,$$

then $C$ is a sextic of torus type.

The statement is suggested by the preceding result that, when $\rho(5)\geq 7$, the map $\sigma_5$ is injective, together with the classification of rank-$19$ sextics. Its resolution status is not specified in the supplied text.