A deterministic Hölder exponent for smooth-transport parabolic SPDEs

On $\mathbb{T}^d$, $d \ge 2$, consider the scalar Itô equation $$du = \nabla \cdot (a \nabla u)\,dt + \sum_{k \ge 1} (b_k \cdot \nabla u)\,dW^k_t, \qquad u(0) = u_0,$$ where the Brownian motions $W^k$ are independent, $u_0 \in C^\infty(\mathbb{T}^d)$ is deterministic, and the coefficients are progressively measurable in $(t, \omega)$ and measurable in $x$, with deterministic bounds $$|a| + \Bigl(\sum_k |b_k|^2\Bigr)^{1/2} \le M, \qquad \eta^\top \Bigl(a - \tfrac12 \sum_k b_k \otimes b_k\Bigr) \eta \ge \nu |\eta|^2 \quad (\eta \in \mathbb{R}^d)$$ for some $M < \infty$ and $\nu > 0$. The solution is the variational one, in $C([0,T]; L^2) \cap L^2(0,T; H^1)$, defined by testing against smooth spatial functions, with the stochastic terms integrated in the Itô sense. Assume moreover that $b = (b_k)_k$ is smooth in $x$ with deterministic bounds $\sup_{t, \omega} \|b(t, \omega, \cdot)\|_{C^j(\mathbb{T}^d; \ell^2(\mathbb{R}^d))} \le M_j$ for every integer $j \ge 1$, while $a$ remains merely bounded and measurable. $C^{\gamma/2, \gamma}$ is the parabolic Hölder space, with exponent $\gamma$ in space and $\gamma/2$ in time.

Agresti, Sauerbrey and Veraar proved that, already when $b$ is bounded in $C^{3+\delta}$, the solution is almost surely Hölder continuous, with an exponent that may depend on $\omega$. Their method, a stochastic change of variables along the flow of the noise, gives an exponent independent of $\omega$ when the $b_k$ depend only on $(t, \omega)$, and for finitely many commuting time-independent fields whose flows satisfy a uniform bound; they also gave an example in two dimensions in which the transformed equation has an ellipticity ratio that exceeds any given bound with positive probability.

Problem. For every $0 < \tau < T$, is there a deterministic $\gamma > 0$, depending only on $d$, $\nu$, $M$, $(M_j)$, $\tau$ and $T$, such that $$\mathbb{P}\bigl(u \in C^{\gamma/2, \gamma}([\tau, T] \times \mathbb{T}^d)\bigr) = 1?$$ The Hölder norm itself may be random and unbounded; the question is whether the exponent can be chosen independently of the sample. No commuting-flow or spatially constant noise assumption is allowed.