Scattering conjecture for global solutions below the mass-energy threshold

Let $V$ satisfy the assumptions $(1.1)$ and $(1.2)$, and let $u(t)$ solve the nonlinear Schrödinger equation with potential $(\textup{NLS}_V)$ and initial data $u_0\in H^1$. Suppose that

$$M[u_0]E[u_0]<\mathcal{ME}$$

and

$$\|u_0\|_{L^2}\|\mathcal{H}^{1/2}u_0\|_{L^2}<\alpha.$$

A global solution satisfying these conditions has finite $S(\dot{H}^{1/2})$-norm, and it scatters in $H^1$.

Scattering conjecture. Every global solution satisfying the conditions above has finite $S(\dot{H}^{1/2})$-norm, and it scatters in $H^1$.

In the homogeneous case, this scattering statement is known, but for the perturbed equation it is posed as a conjecture motivated by the corresponding result of Duyckaerts, Holmer, and Roudenko.