  In holographic inflation, the $4D$ cosmological dynamics is postulated to be
dual to the renormalization group flow of a $3D$ Euclidean conformal field
theory with marginally relevant operators. The scalar potential of the $4D$
theory ---in which inflation is realized--- is highly constrained, with use of
the Hamilton--Jacobi equations. In multi-field holographic realizations of
inflation, fields additional to the inflaton cannot display underdamped
oscillations (that is, their wavefunctions contain no oscillatory phases
independent of the momenta). We show that this result is exact, independent of
the number of fields, the field space geometry and the shape of the
inflationary trajectory followed in multi-field space. In the specific case
where the multi-field trajectory is a straight line or confined to a plane, it
can be understood as the existence of an upper bound on the dynamical masses
$m$ of extra fields of the form $m \leq 3 H / 2$ up to slow roll corrections.
This bound corresponds to the analytic continuation of the well known
Breitenlohner--Freedman bound found in AdS spacetimes in the case when the
masses are approximately constant. The absence of underdamped oscillations
implies that a detection of "cosmological collider" oscillatory patterns in the
non-Gaussian bispectrum would not only rule out single field inflation, but
also holographic inflation or any inflationary model based on the
Hamilton--Jacobi equations. Hence, future observations have the potential to
exclude, at once, an entire class of inflationary theories, regardless of the
details involved in their model building.
