  We consider random gradient fields with disorder where the interaction
potential $V_e$ on an edge $e$ can be expressed as $e^{-V_e(s)} = \int
\rho(\mathrm{d}\kappa)\, e^{-\kappa \xi_e} e^{-\frac{\kappa s^2}{2}}$. Here
$\rho$ denotes a measure with compact support in $(0,\infty)$ and
$\xi_e\in\mathbb{R}$ a nontrivial edge dependent disorder. We show that in
dimension $d=2$ there is a unique shift covariant disordered gradient Gibbs
measure such that the annealed measure is ergodic and has zero tilt. This shows
that the phase transitions known to occur for this class of potential do not
persist to the disordered setting. The proof relies on the connection of the
gradient Gibbs measures to a random conductance model with compact state space,
to which the well known Aizenman-Wehr argument applies.
