Decay conjecture for the wave equation in hyperbolic FLRW universes

Let $\phi$ be a solution of the wave equation in a hyperbolic FLRW universe, with scale factor satisfying the Friedmann equations with zero cosmological constant and equation of state parameter $w$. Assume that the initial data $\phi_0(x)\coloneqq\phi(t_0,x)$ and $\phi_1(x)\coloneqq\partial_t\phi(t_0,x)$ are sufficiently regular and belong to appropriate Sobolev spaces.

Decay conjecture in the hyperbolic case.

$$\|\phi(t,\cdot)\|_{L^{\infty}(\mathbb{H}^3)}\lesssim\begin{cases}(1+t)^{-2},&w\geq\frac13,\\(1+t)^{-\frac{3(w+1)}{2}},&0\leq w\leq\frac13.\end{cases}$$

Moreover, the decay is slower than $(1+t)^{-\frac{3(w+1)}{2}}$ for $-\frac13<w<0$, and there is no decay for $-1\leq w< -\frac13$.

This conjecture describes the expected decay transition for waves on negatively curved FLRW backgrounds. The supplied text motivates it from earlier decay estimates and numerical results, but does not provide a resolution or proof of this hyperbolic case.