The generation conjecture for bichromatic-forbidding 4-candidates

A 4-candidate is the graph class defined in the source context, with four designated external vertices, and a 4-candidate is bichromatic-forbidding when it forbids a proper 4-coloring in which the relevant opposite designated vertices receive the same colors. The diamond is the initial 4-candidate, and the source defines three operations in Lemmas 10ext, extproj, and extprojprime.

4-candidate generation conjecture. Every bichromatic-forbidding 4-candidate can be obtained from the diamond by a sequence of the operations described in Lemmas 10ext, extproj, and extprojprime.

The conjecture is supported by an enumeration of 4-candidates with at most 28 vertices, but the supplied source gives no resolution beyond that computational evidence.