  Given a complete graph $K_{n}=(V, E)$ with non-negative edge costs $c\in
{\mathbb R}^{E}$, the problem $2EC$ is that of finding a 2-edge connected
spanning multi-subgraph of $K_{n}$ of minimum cost. The integrality gap
$\alpha\text{2EC}$ of the linear programming relaxation
$\text{2EC}^{\text{LP}}$ for $2EC$ has been conjectured to be $\frac{6}{5}$,
although currently we only know that
$\frac{6}{5}\leq\alpha\text{2EC}\leq\frac{3}{2}$. In this paper, we explore the
idea of using the structure of solutions for $\text{2EC}^{\text{LP}}$ and the
concept of convex combination to obtain improved bounds for $\alpha\text{2EC}$.
We focus our efforts on a family $J$ of half-integer solutions that appear to
give the largest integrality gap for $\text{2EC}^{\text{LP}}$. We successfully
show that the conjecture $\alpha\text{2EC} = \frac{6}{5}$ is true for any cost
functions optimized by some $x^{*}\in J$.
