Local no-slip Navier–Stokes theory under non-small transport noise

Let $D \subset \mathbb{R}^3$ be a bounded domain with smooth boundary, let $\nu > 0$, and let $b_1, \dots, b_K \in C^\infty(\overline{D}; \mathbb{R}^3)$ be divergence-free vector fields tangent to $\partial D$ that satisfy the stochastic parabolicity condition $$\nu I - \tfrac12 \sum_{k=1}^K b_k \otimes b_k \ge \delta I \quad \text{on } \overline{D}$$ for some $\delta > 0$. Let $\mathbb{P}_D$ be the Helmholtz projection, let $W^1, \dots, W^K$ be independent standard Brownian motions, and consider the Itô equation with the no-slip condition $$du = \bigl[\nu \mathbb{P}_D \Delta u - \mathbb{P}_D \nabla \cdot (u \otimes u)\bigr]\,dt + \sum_{k=1}^K \mathbb{P}_D\bigl[(b_k \cdot \nabla) u\bigr]\,dW^k_t, \qquad u|_{\partial D} = 0,$$ understood in the weak solenoidal $W^{-1,4}$ space, or equivalently through the extrapolated Dirichlet Stokes semigroup. In two dimensions, on an arbitrary open set with the no-slip condition, Agresti and Veraar proved global well-posedness in $L^2$ under the analogous parabolicity condition. On $\mathbb{R}^3$ they proved local well-posedness in critical Besov spaces for transport noise that satisfies the parabolicity condition and is nearly constant at infinity, with no smallness assumption. They observe that for transport noise small in $L^\infty$, the stochastic maximal regularity needed on a domain with the no-slip condition follows from that of the Stokes operator by a perturbation argument.

Problem. For every smooth divergence-free $u_0$ vanishing on $\partial D$, does the equation have a pathwise unique local adapted solution $$u \in C\bigl([0,\tau]; B^{1/2}_{4,4}(D; \mathbb{R}^3)\bigr) \cap L^4\bigl(0,\tau; W^{1,4}_0(D; \mathbb{R}^3)\bigr)$$ for an almost surely positive stopping time $\tau$, with continuous dependence in probability after common stopping? Only the parabolicity condition above is assumed on the noise amplitude, not a small norm of $b$.