Can any finite lattice with at most six elements be realized as an intermediate subfactor lattice?

Question

The paper Lattices of Intermediate Subfactors of Y. Watatani, received on December 1994, finishes by:

Prop. 6.2. $ \ $ Any finite lattice with at most five elements can be realized as an intermediate subfactor lattice.

In fact he has investigated all the lattices with at most six elements, and they can be realized as an intermediate subfactor lattice, except the following two lattices for which he didn't know:

Question: Can any finite lattice with at most six elements be realized as an intermediate subfactor lattice?
In others words: Can $L_{19}$ and $L_{20}$ be realized as intermediate subfactor lattices?

Today is 20 years after this paper of Y. Watatani, and perhaps subfactors realizing these lattices has been found or perhaps we now know how to prove they don't exist.
Of course, if they exist we should ask the same question for seven elements, eight elements... and finally:
Can any finite lattice be realized as an intermediate subfactor lattice?
We've sketched a planar algebraic approach for this question in the optional part here, but we don't know if the skein theory is practicable or not in these cases.

Answer 1

Yes, any finite lattice with at most six elements can be realized as an intermediate subfactor lattice.

The lattice $L_{19}$ and $L_{20}$ are realized as group-subgroup subfactors. This is proved in the proposition 1 of the paper On intervals in subgroup lattices of finite groups (2008) of Michael Aschbacher.

Thanks to Mikko Korhonen who indicates me this paper here.

Remark: It's today a famous open problem to know whether or not any finite lattice can be realized as an intermediate subgroup lattice, see here.

Answer 2

For the record, although Aschbacher's construction of a group with a hexagonal interval is an ingenious construction in finite group theory, involving a rather complicated twisted wreath product, it turns out that his is not the first discovery of such an interval in a subgroup lattice.

Peter Palfy apparently already knew about such hexagons, though their existence did not seem to interest him at the time. (In the paper where I first noticed one, he doesn't even mention its existence.)

On page 477 of the paper On Feit's examples of intervals in subgroup lattices, Journal of Algebra, 116 (2), 1988 (two decades before Aschbacher's example), you will see the hexagonal interval in the subgroup lattice of $A_{11}$ shown below.