This question is related to Can the minimal index of a subfactor take all values in {4cos^2(pi/n);n=3,4,5,...} u [4,infinity]?

I was wondering what one knows for the special case of conformal nets let's say on the circle. For a representation $\pi$ of a conformal net $I\mapsto \mathcal A(I)$ one has a index for the inclusion of type $III_1$ factors: $$\pi(\mathcal A(I)) \subset \mathcal \pi(\mathcal A(I'))'$$ where $I$ is any "proper" intervall on the circle. For the vacuum representation the index is 1 because the inclusion is trivial by Haag duality.

I found that Wassermann showed that the inclusion of $\pi(L_ISU(2)) \subset \pi(L_{I'}SU(2))'$ of positive energy representations at level $\ell$ have index values $\lbrace \sin^2(k \pi/\ell)/ \sin^2(\pi/\ell) \rbrace$. This set contains $4 \cdot \cos^2(\pi/\ell)$ e.g. $k=2$. (btw. I am still looking for the original reference).

Question: Which values can the index take in the set $[4,\infty]$.