Does omitted variable bias affects coefficients for those variables that are not correlated with the error term? (When their is one variable that is)

Question

Does omitted variable bias affects coefficients for those variables that are not correlated with the error term? (When there is one variable that is.) I found two answers, but they appear to be contradictory, although I find both of their arguments convincing. Anyone can clarify this?

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.627.708&rep=rep1&type=pdf pdf page 13 (or manuscript page 6): answers "Yes."

https://ipcig.org/evaluation/apoio/Wooldridge%20-%20Cross-section%20and%20Panel%20Data.pdf pdf page 79 (or manuscript page 62): answers "No."

Answer 1

The answer is "yes" as one can see by constructing a numerical example in Stata. In this example, x3 is the omitted variable which is correlated with x2, but not x1. However, the correlation between x1 & x2 makes the coefficient of x1 biased. Nonetheless, it would be nice if there was a formula for the bias in x1. set obs 10000 generate u1 = runiform() generate u2 = runiform() generate u3 = runiform() generate u4 = runiform() generate u5 = runiform() gen x1 = 0.5u1 + 0.5u4 gen x2 = 0.5u1+0.5u2 gen x3 = 0.5u2 + 0.5u3 gen y = x1 + x2 + x3 + u5 reg y x1 x2