  In the eternal domination game, an attacker attacks a vertex at each turn and
a team of guards must move a guard to the attacked vertex to defend it. The
guards may only move to adjacent vertices and no more than one guard may occupy
a vertex. The goal is to determine the eternal domination number of a graph
which is the minimum number of guards required to defend the graph against an
infinite sequence of attacks. In this paper, we continue the study of the
eternal domination game on strong grids. Cartesian grids have been vastly
studied with tight bounds for small grids such as $2\times n$, $3\times n$,
$4\times n$, and $5\times n$ grids, and recently it was proven in [Lamprou et
al., CIAC 2017, 393-404] that the eternal domination number of these grids in
general is within $O(m+n)$ of their domination number which lower bounds the
eternal domination number. Recently, Finbow et al. proved that the eternal
domination number of strong grids is upper bounded by $\frac{mn}{6}+O(m+n)$. We
adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the
eternal domination number of strong grids is upper bounded by
$\frac{mn}{7}+O(m+n)$. While this does not improve upon a recently announced
bound of $\lceil\frac{m}{3}\rceil \lceil\frac{n}{3}\rceil+O(m\sqrt{n})$ [Mc
Inerney, Nisse, P\'erennes, CIAC 2019] in the general case, we show that our
bound is an improvement in the case where the smaller of the two dimensions is
at most $6179$.
