Linear models — minimum mean squared error against least squares

Consider the simple linear regression model $Y_i=\beta x_i+\epsilon_i$, $i=1,2,…,n$, where $\beta>0$, $\sum_{i=1}^{n}x_i^2>0$, and the uncorrelated errors $\epsilon_i$ have zero mean and finite variance $\sigma^2(>0)$. Let $\tilde\beta_1=\sum_{i=1}^{n}a_i^{*}Y_i$, where the $a_i^{*}$ minimize $\mathbb{E}\left(\sum_{i=1}^{n}a_iY_i-\beta\right)^2$ with respect to scalars $a_1,a_2,…,a_n$. Let $\tilde\beta_2$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?