The six-fifths conjecture for 2-edge-connected multigraphs

Let $G=(V,E)$ be a graph, let

$$\mathrm{LP}(G)=\left\{x\in[0,2]^E:x(\delta(S))\geq 2\text{ for every }\emptyset\subset S\subset V\right\},$$

and let a vector dominate a convex combination of 2-edge-connected multigraphs when it is componentwise at least the incidence vector of that convex combination.

Six-fifths conjecture. If $x\in\mathrm{LP}(G)$, then $\frac{6}{5}x$ dominates a convex combination of 2-edge-connected multigraphs of $G$. Equivalently,

$$\alpha^{\mathrm{LP}}_{\mathrm{2EC}}\leq \frac{6}{5}.$$

This conjecture is stated to be wide open. The paper records only $\frac{6}{5}\leq\alpha^{\mathrm{LP}}_{\mathrm{2EC}}\leq\frac{3}{2}$ in general, with progress for special classes such as half-integer points.