Let $X_t$ be an $\mathbb{R}^d$-valued stochastic process defined on a stochastic base $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ with $\mathbb{P}$-a.s. continuous paths. Suppose that $Z_t$ a local-martingale defined on the same stochastic base.
If $X_t$ and $Z_t$ are equal $\mathbb{P}\otimes m$-a.e. (where $m$ is the Lebesgue measure on $[0,\infty)$), can we conclude that $Z_t$ is a modification of $X_t$?
I'll provide a simple counterexample. Take process $X_t=0$ for $t\in[0,1]$ and $\xi\sim Unif\{-1, 1\}$.
Define process $$ Z_t = \begin{cases} 0&t\in [0,1] \\ \xi&t=1 \end{cases} $$ All the conditions are satisfied but $Z$ is not a modification of $X$, as $P(X_1=Z_1) = 0 \neq 1$.