  We investigate the hierarchy of conic inner approximations
$\mathcal{K}^{(r)}_n$ ($r\in \mathbb{N}$) for the copositive cone
$\text{COP}_n$, introduced by Parrilo (Structured Semidefinite Programs and
Semialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis,
California Institute of Technology, 2001). It is known that
$\text{COP}_4=\mathcal{K}^{(0)}_4$ and that, while the union of the cones
$\mathcal{K}^{(r)}_n$ covers the interior of $\text{COP}_n$, it does not cover
the full cone $\text{COP}_n$ if $n\geq 6$. Here we investigate the remaining
case $n=5$, where all extreme rays have been fully characterized by Hildebrand
(The extreme rays of the 5 $\times$ 5 copositive cone. Linear Algebra and its
Applications, 437(7):1538--1547, 2012). We show that the Horn matrix $H$ and
its positive diagonal scalings play an exceptional role among the extreme rays
of $\text{COP}_5$. We show that equality $\text{COP}_5=\bigcup_{r\geq 0}
\mathcal{K}^{(r)}_5$ holds if and only if any positive diagonal scaling of $H$
belongs to $\mathcal{K}^{(r)}_5$ for some $r\in \mathbb{N}$. As a main
ingredient for the proof, we introduce new Lasserre-type conic inner
approximations for $\text{COP}_n$, based on sums of squares of polynomials. We
show their links to the cones $\mathcal{K}^{(r)}_n$, and we use an optimization
approach that permits to exploit finite convergence results on Lasserre
hierarchy to show membership in the new cones.
