The conjecture on symmetric six-tuples solving Problem 1

A six-tuple $(a,b,c,a,b,c)$ is called symmetric when its first three entries are repeated in the same order in the last three entries. Problem 1 asks which six-tuples have the required property described in the paper.

Symmetric six-tuple conjecture. All symmetric six-tuples $(a,b,c,a,b,c)$ are solutions of Problem 1.

This conjecture is stated after the proof of the relevant obstruction-theoretic cases; the supplied text gives no further resolution or context.