Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable? Suppose we have a model that is:

$$
y = X_1\beta_1 + X_2\beta_2 + X_3\beta_3 + \varepsilon
$$

where $X_1$ is independent of $X_2$ and $X_3$ but $X_2$ and $X_3$ are correlated with each other.

Suppose we don't have data on $X_3$ so we omit it and run the regression:

$$
y = X_1\gamma_1 + X_2\gamma_2 + \nu
$$

Econometric theory on OVB states that $\gamma_2$ will be biased.

My questions:  


Will $\gamma_1$ be biased in this case as well? I ran a simulation with some data I created and I do get a biased result.  
If it is unbiased then how do I prove it? I use FWL theorem to find $\gamma_1$ and show that its expectation equals $\beta_1$ but get stuck here:


$$
\hat{\gamma_1} = \beta_1 + (X_1'M_{X_2}X_1)^{-1}(X_1'M_{X_2}X_3\beta_3) + (X_1'M_{X_2}X_1)^{-1}(X_1'M_{X_2}\varepsilon)
$$

The third term will become 0 when we take the expectation but not sure how to make the second term 0.
I would suspect that if  $X_1$ is independent of $X_3$, then it must imply $E[X_1*X_3]$ is equal to $E[X1] * E[X_3]$, given this you can conclude that the Covariance is 0. $Cov(X_1,X_3) = E[X_1*X_3] - E[X_1]* E[X_3]$. If the Covariance is 0 between these two variables then so is the correlation. Thus I would believe that removing $X_3$ from the regression and having it's effect enter the error term would not result in having $γ_1$ be biased as the $X_3$ in the residual term does not have correlation with $γ_1$. 

Are you looking for a rigorous mathematical proof?