The constructive generation conjecture for efficient total girth colorings

Let $G$ be a finite connected simple cubic graph of girth $4$, and let an ETGC denote an efficient total girth coloring. The six constructive tools considered in the source are sprays, extensions, unfoldings, exchanges, amalgams, and piercing; sprays yield the smallest such graph, $G=Q_3$. Constructive generation conjecture. ETGCs of finite connected simple cubic graphs of girth $4$ are obtained solely by means of these six constructive tools. The supplied text does not establish whether this proposed completeness statement is proved or remains open.