  On exponentially expanding Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW)
spacetimes, there is a distinguished family of spatially homogeneous and
isotropic solutions to the relativistic Euler equations with a linear equation
of state of the form $p=\sigma \rho$, where $\sigma \in [0,1]$ is the square of
the sound speed. Restricting these solutions to a constant time hypersurface
yields initial data that uniquely generates them. In this article, we show, for
sound speeds satisfying $\frac{1}{3}<\sigma <\frac{k+1}{3k}$ with $k\in
\mathbb{Z}_{>\frac{3}{2}}$, that $\mathbb{T}^2$-symmetric initial data that is
chosen sufficiently close to spatially homogeneous and isotropic data uniquely
generates a $\mathbb{T}^2$-symmetric solution of the relativistic Euler
equations that exists globally to the future. Moreover, provided $k\in
\mathbb{Z}_{>\frac{5}{2}}$, we show that there exist open sets of
$\mathbb{T}^2$-symmetric initial data for which the fractional density gradient
becomes unbounded at timelike infinity. This rigorously confirms, in the
restricted setting of relativistic fluids on exponentially expanding FLRW
spacetimes, the fractional density gradient blow-up scenario conjectured by
Rendall in \cite{Rendall:2004}.
