  Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact
manifold $(X, g)$ and consider the zeta-regularized trace
$\mathrm{Tr}_\zeta(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$,
then the supremum of this quantity, taken over all metrics $g$ of fixed volume
in the conformal class, is always greater than or equal to the corresponding
quantity on the standard sphere. Moreover, we show that in the case that it is
strictly larger, the supremum is attained by a metric of constant mass. Using
positive mass theorems, we give some geometric conditions for this to happen.
