Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable?
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?