ag.algebraic geometry - Pole of a function in a diagonal embedding of a normal affine variety.

Let $X$ be a normal affine algebraic variety of dimension $n$ and ${bar X}^1$, ${bar X}^2$ be complete normal varieties containing $X$ as a (Zariski) dense open subset. Let $f$ be a regular function on $X$ such that $f$ has a pole along every irreducible component (which is necessarily of dimension $n-1$) of $bar X^i setminus X$ for each $i$, $1 leq i leq 2$. Let $bar X subseteq {bar X}^1 times {bar X}^2$ be the closure of the diagonal embedding of $X$ into ${bar X}^1 times {bar X}^2$. Is it true (assuming that $bar X$ is also normal) that $f$ has a pole along every irreducible component of $bar Xsetminus X$?

It is easy to see that this is true when $n=2$ or when $X$ is a (Zariski) open subset of a torus $mathbb T$ and ${bar X}^i$'s are toric completions of $mathbb T$. But the general case still eludes me.