= Solution
Write $\mu_j=\delta F/\delta p_j$ and $\int=\int_{t_1}^{t_2}dt\int d\mathbf r$. Eliminating the noise gives $f=\dot p+\Gamma\mu$. The <Onsager--Machlup path probability> is therefore
$$
\boxed{P_F[p]=\mathcal K[p]\exp\left[-\frac1{2\sigma^2}\int|\dot p+\Gamma\mu|^2\right],\qquad
P_B[p]=\mathcal K[p]\exp\left[-\frac1{2\sigma^2}\int|-\dot p+\Gamma\mu|^2\right].}
$$
These are conditional path densities given the corresponding starting configuration. For a time-even <vector order parameter>, reversal reads the configurations backward, changes $\dot p$ to $-\dot p$, and keeps $\mu$ unchanged on each corresponding configuration.
The two <Gaussian white noise> measures have the same normalization. With a common midpoint discretization, the noise-to-path Jacobian also agrees under reversal by <time-reversal invariance of a path Jacobian>. The common factor $\mathcal K[p]$ may include that path-dependent Jacobian; it need not be a universal constant. This makes the shared-factor assertion precise while leaving the requested ratio unaffected.
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