  It is expected that, under mild conditions, local Langlands correspondence
preserves depths of representations. In this article, we formulate a
conjectural geometrisation of this expectation. We prove half of this
conjecture by showing that the depth of a categorical representation of the
loop group is less than or equal to the depth of its underlying geometric
Langlands parameter. A key ingredient of our proof is a new definition of the
slope of a meromorphic connection, a definition which uses opers. In the
appendix, we consider a relationship between our conjecture and Zhu's
conjecture on non-vanishing of the Hecke eigensheaves produced by Beilinson and
Drinfeld's quantisation of Hitchin's fibration for non-constant groups.
