Diamond diagrams and multivariable $(\varphi,\mathcal{O}_K^{\times})$-modules

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. Let $\pi$ be an admissible smooth mod $p$ representation of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspaces of the mod $p$ cohomology and $\overline{r}$ be its underlying global two-dimensional Galois representation. When $\overline{r}$ satisfies some Taylor-Wiles hypotheses and is sufficiently generic at $p$, we compute explicitly certain constants appearing in the diagram associated to $\pi$, generalizing the results of Dotto-Le. As a result, we prove that the associated \'etale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil-Herzig-Hu-Morra-Schraen is explicitly determined by the restriction of $\overline{r}$ to the decomposition group at $p$, generalizing the results of Breuil-Herzig-Hu-Morra-Schraen and the author.