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.