Strictness conjecture for the filtration of classes $\mathfrak L_\alpha$

Let $\{\mathfrak L_\alpha\}_{\alpha\in[0,1]}$ be the filtration of $\mathcal G_0$ defined in the paper. The family is called strict when its inclusions are strict as the parameter increases. Strictness conjecture. The filtration

$$\{\mathfrak L_\alpha\}_{\alpha\in[0,1]}$$

is strict. The paper has already established that this family is a filtration and that its boundaries are nonempty for $\alpha>\frac12$; the supplied text does not establish strictness, so the assertion remains open.