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 $C^{\infty}$ 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 $\gamma_h'(0)=h$ satisfies $\lim_{t \to \infty}$ $R(\gamma_h(t))=-\infty$, uniformly.