Dejter's four-operation conjecture for efficient total colorings of cubic graphs

Let $\Gamma$ be a finite connected simple cubic graph of girth $4$. An efficient total coloring (ETC) is a total coloring in which each color class induces an efficient dominating set on the vertex set. The cube $Q_3$ is the 3-dimensional cube graph.

Dejter's four-operation conjecture. Every ETC of $\Gamma$ is obtained via the iterative genus-increasing procedure of Corollary~, departing from $Q_3$, through four constructive operations: periodic extensions, accordion unfoldings, cycle exchanges, and ETCings.

The conjecture proposes a complete construction of efficient total colorings for finite connected simple cubic graphs of girth $4$. The source gives the construction framework but no resolution of the claim.