Avoidance conjecture for fractional colorings of 4-cliques

Let $G$ be a graph with maximum degree $5$ and clique number $4$, such that no two $4$-cliques intersect and no vertex outside any maximum clique $C$ has more than one neighbour in $C$. Avoidance conjecture. There is a fractional $4$-colouring of the vertices in $4$-cliques such that, for every vertex $v$ not in a $4$-clique,

$$|\alpha(v)|\geq 1.$$

The paper presents this as a further conjecture intended to improve its bounds for the cases involving maximum degree $5$ and $6$.