posets – Which Boolean lattices have a left-to-right symmetric drawing?

This question is inspired by a similar MSE question about partition lattices.

Question: Which finite Boolean lattices have a symmetric drawing on the 2D plane?

By a symmetric drawing of a lattice, I mean the usual Hasse diagram, with the elements as vertices and the cover relation shown as upward edges, with the following conditions:

  • no two elements are drawn in the same location,
  • no edge $uv$ is drawn through some other element $w$,
  • the drawing is symmetric about the $y$ axis (considering the vertices as points and edges as lines, either straight or curved — but if curved, the curves must obey the symmetry).

To set the stage, we observe that $B_1$ to $B_4$ have symmetric drawings. $B_3$ is well-known, but $B_4$ is not obvious. Note that although Boolean lattices are graded, we allow elements of the same rank to be drawn at different heights (14 and 23 on the right).

Symmetric drawings of B3 and B4

Can we at least settle the question for $B_5$?

A more general question (answers to this would also be welcome): Is there an efficient method of deciding whether a given lattice admits a symmetric drawing, and (possibly) producing such a drawing?