= Solution
Put $u_j=\dot p_j+\partial_iW_{ij}$. The noises required by a joint trajectory are $f_j=u_j+\Gamma\mu_j$ and $N_{ij}=W_{ij}+M\partial_i\mu_j$. Their independent <Gaussian white noise> weights give the <joint path probability of an order parameter and its flux>:
$$
\boxed{P_F[p,W]=\mathcal K[p,W]\exp\left[-\int\left\{\frac{|u+\Gamma\mu|^2}{2\sigma^2}+\frac{|W+M\nabla\mu|^2}{2\sigma_N^2}\right\}\right].}
$$
For the physically reversed path, $p$ is time-even and the transport flux $W$ is time-odd. Therefore $u\mapsto-u$, $W\mapsto-W$, and
$$
\boxed{P_B[p,W]=\mathcal K[p,W]\exp\left[-\int\left\{\frac{|-u+\Gamma\mu|^2}{2\sigma^2}+\frac{|-W+M\nabla\mu|^2}{2\sigma_N^2}\right\}\right].}
$$
The common factor includes the two noise normalizations and the midpoint <Jacobian determinant>. Its calculation must include both relaxation channels. On a finite spatial grid, let $\mathsf D$ and $\mathsf G$ represent the <divergence> and <gradient>, with $\Delta=\mathsf D\mathsf G$ the discrete <Laplacian>, and let $\mathcal H_n=\partial\mu/\partial p$ be the <Hessian matrix> of $F$ at the midpoint of time step $n$. Differentiating the two required noises with respect to the next configuration and the interval flux gives the <Jacobian matrix>
$$
\frac{\partial(f_n,N_n)}{\partial(p_{n+1},W_n)}=
\begin{pmatrix}
\frac{I}{\Delta t}+\Gamma\mathcal H_n/2&\mathsf D\\
M\mathsf G\mathcal H_n/2&I
\end{pmatrix}.
$$
Using the <Schur complement>, its <Jacobian determinant>, apart from the path-independent power of $\Delta t$, is
$$
\det\left[I+\frac{\Delta t}{2}(\Gamma I-M\Delta)\mathcal H_n\right].
$$
Reversal visits the same midpoint configurations in reverse order, so the product of these factors is unchanged. The joint <time-reversal invariance of a path Jacobian> therefore involves the complete relaxation operator $\Gamma I-M\Delta$.
Taking the action difference gives
$$
\boxed{\log\frac{P_F}{P_B}=-\frac{2\Gamma}{\sigma^2}\int\mu_j(\dot p_j+\partial_iW_{ij})-\frac{2M}{\sigma_N^2}\int W_{ij}\partial_i\mu_j.}
$$
Repeated component indices are summed. The second squared norm sums over both flux indices. This is <mixed conserved and nonconserved order-parameter dynamics>; reversing the order parameter history while leaving the flux unreversed would give the wrong probability ratio.
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