Suppose I want to estimate the model
$Y_i = \beta_0 + \beta_1 D_i + \beta_2 X_i + \beta_3 D_i \times X_i + u_i$,
where $D_i$ is a dummy variable and $X_i$ a continuous variable. Furthermore, assume $corr(D_i, u_i)=0$ and $corr(X_i, u_i)\neq 0$. Can we use OLS to get an unbiased estimate of $\beta_1$ and $\beta_3$? What is the underlying proof and intuition?