Let $G$ be a connected reductive group over a local non-archimedean field $K$. Let $\widehat{K}^{nr}$ be the completion of the maximal unramified extension of $K$ and let $\sigma$ denote the Frobenius automorphism of $\widehat{K}^{nr}/K$. We have the $\sigma$-conjugacy classes in $G(\widehat{K}^{nr})$, that is subsets of the form $[b] = \{g^{-1}b\sigma(g) \colon g \in G(\widehat{K}^{nr})\}$; more specifically, there is the notion of basic $\sigma$-conjugacy classes. These we studied by Kottwitz (Isocrystals with additional structure, Isocrystals with additional structure II).

My question is now:

1) Is it true that any $\sigma$-conjugacy class has a representative in $G(K)$?

(surely, the answer is yes for G = GL_n). If the answer to question 1) is no, I am still interested in the following specific variants of this question:

2) Does the answer get "yes", if we restrict attention to (a) basic $\sigma$-conjugacy classes, or (b) unramified (= split $\widehat{K}^{nr}$ + quasi-split over $K$) group $G$, or (c) both, basic $\sigma$-conjugacy classes and unramified $G$.