The Goldberg snark circular-flow-number conjecture

For every positive integer $k$, let $G_{2k+1}$ be the Goldberg snark on $8(2k+1)$ vertices, and let $\Phi_c(G_{2k+1})$ denote its circular flow number. The Goldberg snark circular-flow-number conjecture.

$$\Phi_c(G_{2k+1})=4+\frac{1}{k+1}.$$

The paper verifies this equality for $G_3$, $G_5$, and $G_7$, while proving the corresponding upper bound for every positive integer $k$. Optimality of that upper bound remains open.