The eventual fractional-defect conjecture for connected 4-regular graphs

A graph $G$ is $4$-regular if every vertex has degree $4$. For a graph $G$, let $D(G,2)$ denote the minimum defect in a fractional $2$-coloring. The exceptional graphs are the compositions $C_m[2K_1]$ of odd cycles with the independent graph on two vertices. Eventual fractional-defect conjecture for connected $4$-regular graphs. Apart from $G=C_m[2K_1]$ where $m$ is odd, it holds that

$$D(G)<2$$

for all but finitely many connected $4$-regular graphs.

Lovász's result gives the general upper bound $D(G,2)\le 2$ for $4$-regular graphs. The exceptional odd-cycle compositions attain $D(G,2)=2$, while computational searches support strict inequality for many other graphs; the conjecture asserts that only finitely many connected exceptions remain beyond those compositions.