Order-detection converse for left-orderable gluings

Let $M_1$ and $M_2$ be knot manifolds, and let

$$W=M_1\cup_f M_2,$$

where $f:\partial M_1\xrightarrow{\;\cong\;}\partial M_2$ is a homeomorphism. A slope $[\alpha_i]$ belongs to the set $\mathcal{D}_{ord}(M_i)$ of order-detected slopes on $\partial M_i$. Order-gluing conjecture. If $\pi_1(W)$ is left-orderable, then $f$ identifies slopes $[\alpha_1]\in\mathcal{D}_{ord}(M_1)$ and $[\alpha_2]\in\mathcal{D}_{ord}(M_2)$. This is the expected converse to the paper's sufficient gluing criterion; it has been verified in some cases, including when every left-order on one factor is boundary-cofinal.