  We consider Helmholtz problems with a perturbed wave speed, where the
single-signed perturbation is linear in a parameter $z$. Both the wave speed
and the perturbation are allowed to be discontinuous (modelling a penetrable
obstacle). We show that there exists a polynomial function of frequency such
that, for any frequency, for most values of $z$, the norm of the solution
operator is bounded by that function.
  This solution-operator bound is most interesting for Helmholtz problems with
strong trapping; recall that here there exists a sequence of real frequencies,
tending to infinity, through which the solution operator grows
superalgebraically, with these frequencies often called
$\textit{quasi-resonances}$. The result of this paper then shows that, at every
fixed frequency in the quasi-resonance, the norm of the solution operator
becomes much smaller for most single-signed perturbations of the wave speed,
i.e., quasi-resonances are unstable under most such perturbations.
