= Solution
Let $\delta=C-\mu>0$, and first suppose $\sigma>0$. For the infinite-horizon contraction, use the usual <sublinear path space for fluid queues>,
$$
\mathcal C_0=\{f\in C([0,\infty)):f(0)=0,\ f(t)/(1+t)\to0\},
\qquad \|f\|_{\rm sl}=\sup_{t\geq0}\frac{|f(t)|}{1+t}.
$$
We interpret the supplied path-space <large deviation principle> in this norm, or in a topology with the same workload-continuity property. The paper does not specify its norm explicitly; mere convergence on compact time intervals would not justify an infinite-horizon contraction.
Define the <queue workload> functional
$$
R(f)=\sup_{t\geq0}\{\sigma f(t)-\delta t\}.
$$
It is finite and nonnegative on the <sublinear path space for fluid queues>. Here is the required continuity argument. Fix $f$ and choose $T\geq1$ so that $|\sigma f(t)|\leq\delta t/4$ for $t\geq T$. If $\|g-f\|_{\rm sl}<\delta/(4\sigma)$, then for $t\geq T$,
$$
\sigma g(t)-\delta t
\leq\frac{\delta t}{4}+\frac{\delta(1+t)}4-\delta t
=\frac{\delta}{4}-\frac{\delta t}{2}<0.
$$
The corresponding expression for $f$ is also negative there, whereas at zero both are zero. Each <supremum> can therefore be restricted to $[0,T]$, giving
$$
|R(g)-R(f)|\leq\sigma(1+T)\|g-f\|_{\rm sl}.
$$
Thus $R$ is a <continuous map>; this proof also shows why the negative drift is essential.
Substituting $t=Ns$ into the <queue workload> gives
$$
\frac{r(X)}N=\sup_{s\geq0}\left\{\sigma\frac{Z(Ns)}N-\delta s\right\}
=R\!\left(\frac{Z(N\,\cdot)}N\right).
$$
The <contraction principle for large deviations> says that a continuous image of a family with <good rate function> $I$ has the same <large-deviation speed> and <good rate function> obtained by minimizing $I$ over each fibre. Consequently
$$
\boxed{\text{speed }N^{2(1-H)},\qquad
J(q)=\inf_{\{f:R(f)=q\}}I(f)\quad(q\geq0),\qquad J(q)=+\infty\quad(q<0).}
$$
In particular $J(0)=0$: $Z/\sqrt L\to0$ in the path norm almost surely, so the given <large deviation principle> and <lower semicontinuity> give $I(0)=0$, and $R(0)=0$. If $\sigma=0$, the <queue workload> is identically zero and the <rate function> is zero at zero and infinite elsewhere. No unspecified <variance> normalization of the <fractional Brownian motion> is needed for this variational answer.
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