Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{ 0, 1, . . ., k\}$ is called a \emph{ $\{ k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{ k\}$-dominating function is the sum of its values over all the vertices. The $\{ k\}$-domination number of $G$, $\gamma_{ \{ k\}}(G)$, is defined to be the minimum weight taken over all $\{ k\}$-domination functions. Bre\v{ s}ar, Henning, and Klav\v{ z}ar (On integer domination in graphs and Vizing-like problems. \emph{ Taiwanese J. Math.} { 10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $\gamma_{ \{ k\}}(G\square H)\ge \gamma(G)\gamma(H)$. In this note we use the Roman $\{ 2\}$-domination number, $\gamma_{ R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{ 2\}$-domination. \emph{ Discrete Applied Mathematics} { 204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $\gamma_{ \{ 2\}}(G\square H)\ge \gamma_{ R2}(G\square H)\ge \gamma(G)\gamma(H)$, which also implies the conjecture for all $k\ge 2$.
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