Consider
$$(\hat\beta_0, \hat\beta) = \arg\min_{\beta_0,\beta}\|y-\beta_0 1 - X\beta\|^2$$
and
$$(\tilde\beta_0, \tilde\beta) =\arg\min_{\beta_0,\beta} = \left\|y- \left( \beta_0 - \frac{1^T X \beta}{n}\right) 1 - X\beta \right\|^2 $$
Note that $1, y \in \mathbb{R}^{n \times 1}$ and $X \in \mathbb{R}^{n \times p}$ and $\beta \in \mathbb{R}^{p \times 1}$
Is it true that $\hat\beta_0 = \tilde\beta_0 - \frac{1^T X \tilde\beta}{n}$ and $\hat\beta = \tilde\beta$?