The outerplanar graph fractional-defect conjecture

A $2$-coloring assigns each vertex a red usage in $[0,1]$, with blue usage $1-r(v)$; the defect of a vertex is the sum, over its neighbors, of the overlap in their color usages, and $D(G,2)$ is the minimum possible maximum defect over all such colorings of $G$. Outerplanar graph fractional-defect conjecture. For any outerplanar graph $G$,

$$D(G,2)<2.$$

Ordinary $2$-colorings of outerplanar graphs have defect at most $2$, and the fan examples show that the minimum defect can approach $2$ from below. The conjecture asks whether the bound is always strictly improvable.