Abstract: According to a Conjecture of Fargues and Scholze the local Langlands correspondence for GL_n(Q_p) has an upgrade to an equivalence of categories between the (derived) category of certain sheaves on the stack of n-dimensional vector bundles on the so-called Fargues-Fontaine curve and the (derived) category of coherent sheaves on the stack of n-dimensional Weil-Deligne representations. I will report on a joint project with Arthur Bartles where we give evidence to this conjecture by proving that both categories have the same K-theory. In this talk I will explain the statements and give an idea how to compute the K-theory on both sides.