Local boundedness conjecture for critical-drift Stokes solutions

Let $B_2\subset\boldsymbol{\mathbb{R}}^3$ be the ball of radius $2$, and let $b\in L^{3,\infty}(B_2)$ be divergence free. Let $u\in W^{1,(\frac{3}{2},\infty)}(B_2)$ be a very weak solution of the perturbed Stokes system with zero force in $B_2$, meaning that

$$\int_{B_2}(\nabla u-b\otimes u):\nabla\zeta=0$$

for every $\zeta\in C^\infty_{c,\sigma}(B_2)$. Then $u$ is essentially bounded on $B_1$.

Local boundedness conjecture. Under these hypotheses, $u\in L^\infty(B_1)$.

This is a local regularity problem for the perturbed Stokes system at the critical drift integrability. It is presented as relevant to the Landau-solution classification problem; the source does not state a resolution.