The enumeration conjecture for 3-minimal graphs

A 3-minimal graph is a graph whose relevant Lovász–Schrijver semidefinite-programming hierarchy obstruction has minimal rank three. Two graphs are considered the same when they are isomorphic.

Enumeration conjecture. There are exactly $49$ non-isomorphic $3$-minimal graphs.

The computational search found 49 such graphs and provides evidence that the displayed list is complete, but the statement is presented as a conjecture rather than an established theorem.