Strengthened lower-bound conjecture for the circular flow number of snarks

Let $G$ be a connected bridgeless cubic graph of order at most $8k+8$ that does not admit any $3$-edge-coloring.

Strengthened circular-flow conjecture.

$$\Phi_c(G) \ge 4+\frac{1}{k}.$$

This conjecture strengthens the lower bound of Lukot'ka and Škoviéra for the circular flow number of non-$3$-edge-colorable cubic graphs. The paper reports computational evidence for it, but no proof or resolution is given.