Local geometric Langlands for the semi-infinite Whittaker category

I'll describe work-in-progress by my graduate student Sam Raskin.
Local geometric Langlands in its most general form is a conjecture that certain two 2-categories (one associated with the group $G$, and another with its Langlands dual) are equivalent. However, the very formulation of this conjecture relies on a series (of largely conjectural) statements that certain pairs of factorization categories are equivalent. The most fundamental among them is a description of that the category of Whittaker D-modules on the semi-infinite flag space $G(K)/N(K)T(O)$ in Langlands dual terms. In the talk I'll explain this description, as well as the role that it plays in the (usual) global unramfied geometric Langlands.