The total form domain intersection conjecture for uniform dynamical decoupling

Let $\mathcal{H}_s$ be a system Hilbert space, let $V$ be a decoupling set for $\mathcal{H}_s$, and let $H$ be a non-negative Hamiltonian on a Hilbert space $\mathcal{H}$. The total form domain intersection is

$$\bigcap_{v\in V} v\mathcal{D}(H^{1/2})\subset\mathcal{H}.$$

Total form domain intersection conjecture. If dynamical decoupling works uniformly, then

$$\bigcap_{v\in V} v\mathcal{D}(H^{1/2})\subset\mathcal{H}$$

must be dense. The preceding necessary condition for uniform dynamical decoupling requires pairwise form-domain intersections to be dense; the conjecture strengthens this to the intersection over all elements of the decoupling set. The source explains that a proof would require a currently open generalisation of Kato's result to Trotter products of arbitrarily many semigroups.