Energy-critical NLS with quadratic-growth potentials global well-posedness conjecture

Let $H=-\tfrac{1}{2}\Delta+V$ be the positive Schrödinger operator associated with a potential $V$ of quadratic growth, let $Q(H)=D(H^{1/2})$ be its form domain, and consider the energy-critical equation

$$(i\partial_t+\tfrac{1}{2}\Delta)u=\mu|u|^{\frac{4}{d-2}}u+V(x)u, \qquad u(0)=u_0\in Q(H).$$

For a compact interval $I\subset\mathbb{R}$, write

$$S_I(u)=\int_I\int_{\mathbb{R}^d}|u(t,x)|^{\frac{2(d+2)}{d-2}}\,dx\,dt.$$

Global well-posedness conjecture. When $\mu=1$, the equation is globally well posed: for each $u_0\in Q(H)$ there is a unique global solution $u:\mathbb{R}\times\mathbb{R}^d\to\mathbb{C}$ with $u(0)=u_0$, and, for every compact interval $I\subset\mathbb{R}$,

$$S_I(u)\leq C(|I|,\|u_0\|_\Sigma).$$

When $\mu=-1$, the same conclusion holds provided

$$E(u_0)<E_\Delta(W),\qquad \|\nabla u_0\|_{L^2}\leq\|\nabla W\|_{L^2}.$$

The conjecture is motivated by the analogous scale-invariant problem, but the quadratic potential has discrete spectrum, so global-in-time spacetime bounds of the free type cannot be expected; the source does not state a resolution for this potential problem.