  We compare two statements of the refined local Langlands correspondence for
connected reductive groups defined over a p-adic field -- one involving
Kottwitz's set B(G) of isocrystals with additional structure, and one involving
the cohomology set H^1(u -> W,Z -> G) introduced in arXiv:1304.3292. We show
that if either statement is valid for all connected reductive groups, then so
is the other. We also discuss how the second statement depends on the choice of
element of H^1(u -> W,Z -> G).
