Unique minimal diagram conjecture for free knots of girth at least four

Let $K$ be a framed four-valent graph, viewed as a free knot diagram, and let its girth be the length of its shortest cycle. If

$$\operatorname{girth}(K)\geq 4,$$

then the unique minimal diagram conjecture. The diagram $K$ is the unique minimal diagram for the knot class of $K$. The preceding discussion observes that free knot diagrams have no loops or bigons, and that a diagram of girth at least four admits neither a decreasing Reidemeister move nor a third Reidemeister move. The conjecture would therefore provide a minimality criterion and a recognition algorithm for a large class of free knots.