I will explain how to prove the Breuil-Mezard conjecture for split (non-scalar) residual representations by local methods. Combined with the cases previously proved by Kisin and Paskunas, this completes the proof of the conjecture for $\mathrm{GL}_2(\mathbb{Q}_p)$. As an application, we can prove the Fontaine-Mazur conjecture in the cases that the global residual representation restricts to the decomposition group at $p$ as an extension of the trivial character by the mod $p$ cyclotomic character. These are the cases complementary to Kisin's. This is a joint work with Yongquan Hu.