Relationship between local time of one dimensional Brownian motion and multiple dimensional Brownian motion

Question

Let $X_t$ be a one-dimensional reflecting Brownian motion with drift $\mu_1$ reflecting above $0$. Let $Y_t$ be a one-dimensional Brownian motion with drift $\mu_2$ independent of $X_t$. Then $Z_t := (X_t, Y_t)$ is a two-dimensional Brownian motion with drift $(\mu_1, \mu_2)$ on the half-plane reflecting above $x=0$. What is the relationship between the local time $\ell_t^0(X)$ of $X_t$ at $0$ and the local time $\ell_t^{x=0}(Z)$ of $Z_t$ at $x=0$? Is it true that because the movement in the $x$ direction of $Z_t$ is driven entirely by $X_t$ these two local times would be equal?