Quadratic embeddability conjecture for theta graphs

Let $\Theta(\alpha,\beta,\gamma)$ be the theta graph formed from three internally vertex-disjoint paths of lengths $\alpha$, $\beta$, and $\gamma$ between two common endpoints, where $1\leq\alpha\leq\beta\leq\gamma$ and at most one of $\alpha,\beta,\gamma$ equals $1$. A graph is of QE class if it has the quadratic embeddability property.

Quadratic embeddability conjecture. Assume $1\leq\alpha\leq\beta\leq\gamma$ and $\beta\geq2$. Then $\Theta(\alpha,\beta,\gamma)$ is of QE class if and only if either

$$\alpha=1,$$

or

$$\alpha=2,\qquad \beta=3,\qquad \gamma\text{ is odd}.$$

The preceding theorems establish the QE and non-QE classifications for several broad families of theta graphs, leaving the stated exceptional case as the remaining classification predicted by the conjecture. The conjecture would give a complete characterization of quadratic embeddability for theta graphs under the indicated ordering and parameter restrictions.