Scalar curvature along Ebin geodesics

Let $M$ be a smooth, compact manifold and let ${\mathcal{N}}_{\mu }$ denote the set of Riemannian metrics on $M$ with smooth volume density $\mu$. For a given $g_0\in {\mathcal{N}}_{\mu }$, we show that if $\dim (M)\ge 5$, then there exists an open and dense subset ${\mathcal{Y}}_{g_0}\subset T_{g_0}{\mathcal{N}}_{\mu }$ (in the $C^{\mathrm{\infty }}$ topology) so that for each $h\in {\mathcal{Y}}_{g_0}$, the $({\mathcal{N}}_{\mu },L^2)$ Ebin geodesic ${\gamma }_h(t)$ with ${\gamma }_h(0)=g_0$ and ${\gamma }_h^{\mathrm{\prime }}(0)=h$ satisfies ${\lim }_{t\to \mathrm{\infty }}$ $R({\gamma }_h(t))=-\mathrm{\infty }$, uniformly.