The closure-equality conjecture for mapping-class-group orbit closures

Let $S$ be a nonorientable surface, and let $\lambda$ be a measured lamination. Consider the inclusions in the two closure theorems referred to in the source.

Closure-equality conjecture. The inclusions in Theorems and are equalities, except when $\lambda$ is the bounding geodesic of the genus-three closed surface.

This is the more general conjecture for which the preceding rational-density conjecture is described as a first step. The notation and the precise inclusions are not supplied in the extracted context, so the exact mathematical content should be checked against the source.