Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois representation. Diamond gave a classification of all such Galois representations into four families $P$, $S$, $V$ and $H$.

I understand that there is no such classification when $\text{GL}_2$ is replaced by $\text{GL}_n$ for $n>2$ or $\text{GSp}_{2n}$ for $n>1$. Does one expect that there should be a generalization of Diamond's classification which has not been discovered yet, or is there a reason why no such description should exist as in the $\text{GL}_2$ case?