  We define a filtration of a standard Whittaker module over a complex
semisimple Lie algebra and and establish its fundamental properties. Our
filtration specialises to the Jantzen filtration of a Verma module for a
certain choice of parameter. We prove that embeddings of standard Whittaker
modules are strict with respect to our filtration, and that the filtration
layers are semisimple. This provides a generalisation of the Jantzen
conjectures to Whittaker modules. We prove these statements in two ways. First,
we give an algebraic proof which compares Whittaker modules to Verma modules
using a functor introduced by Backelin. Second, we give a geometric proof using
mixed twistor $\mathcal{D}$-modules.
