If partial regression coefficient is zero, then $Y$ is independent of $X_i$ conditional on all other regression variables In a textbook Causal Inference in Statistics - A Primer (p. 81), it says

Given the regression equation $$y=r_{0}+r_{1} x_{1}+r_{2}x_{2}+\cdots+r_{n} x_{n}+\epsilon$$
if $r_{i}=0$, then $Y$ is independent of $X_i$ conditional on all the
other regression variables.

I get that if $r_{i}=0$, a correlation between $Y$ and $X_i$ is 0. But how can they be conditionally independent?

Under the condition that all the $X_k$ with $k\ne i$ are fixed, i.e. $X_k = x^\ast_{k}$, we have that $Y$ is given as:
$$
\begin{align}
Y &= r_0 + r_1x^\ast_1 + \ldots + r_{i-1}x^\ast_{i-1} \;+ r_i x_i + \; r_{i+1}x^\ast_{i+1} + \ldots + r_nx^\ast_n + \epsilon\\
    &= f + r_ix_i + \epsilon,
\end{align}
$$
where $f=r_0 + \sum_{k\ne i} r_kx^\ast_k$ is simply a fixed constant. Now, if $r_i=0$, $Y$ reduces to
$$
Y = f + \epsilon,
$$
i.e. the noise $\epsilon$ shifted by $f$ and thus is clearly independent of $X_i$ (I presume that the noise is defined to be independent of all $X_r$).
