The line-finding conjecture for cubic surfaces

Let $\mathcal{H}_{3,3}$ denote the moduli space of smooth cubic surfaces, and let $\mathcal{H}_{3,3}(r)$ denote the moduli space of smooth cubic surfaces together with $r$ specified lines. The resolvent degree $\operatorname{RD}(\mathcal{H}_{3,3}(r)\to\mathcal{H}_{3,3})$ measures the complexity of finding the lines on a general cubic surface. The line-finding conjecture.

$$\operatorname{RD}(\mathcal{H}_{3,3}(27)\to \mathcal{H}_{3,3})=\operatorname{RD}(\mathcal{H}_{3,3}(1)\to \mathcal{H}_{3,3})=3.$$

This asserts that finding all $27$ lines, or even one line, on a general smooth cubic surface has resolvent degree exactly $3$. The surrounding discussion gives the upper bound via classical constructions, while the required lower bound is part of the paper's study of lower bounds for resolvent degree.