One-edge completion of 1-11-representability

Prove, without relying on computational verification, that every graph on eight vertices can be converted into a word-representable graph by adding at most one edge.

Background

Computational experiments reportedly indicate that each of the 929 non-word-representable graphs on eight vertices becomes word-representable after adding a single edge. The paper's proof of 1-11-representability often adds more than one edge, so a noncomputational proof of the one-edge property would provide new techniques and might help address graphs on nine vertices.