This is the handwritten notes of the talk. Before taking this, you are particularly recommended to study this lecture series on mod $p$ Langlands program for $\mathrm{GL}_2$ by Yongquan Hu at BICMR.
Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$, and $\pi$ a smooth representation of $\mathrm{GL}_2(K)$ on some Hecke eigenspace in the $H^1$ mod $p$ of a Shimura curve. One can associate to $\pi$ a multivariable $(\varphi, \mathcal{O}_K^*)$-module $\mathbb{D}_A(\pi)$. I will state a conjecture which describes $\mathbb{D}_A(\pi)$ in terms of the underlying 2-dimensional mod $p$ representation of $\mathrm{Gal}(\overline{K}/K)$. When the latter is semi-simple (sufficiently generic), I will sketch a proof of this conjecture. This is joint work with F. Herzig, Y. Hu, S. Morra and B. Schraen.