Balanced-graph conjecture for flat graphs

Let $G$ be a flat graph, and let $c_i(G)$ denote the number of $i$-cliques in $G$. Define

$$D_{clique}(G)=\left|\sum_{i:\ i\ \mathrm{is\ odd}}c_i(G)-\sum_{i:\ i\ \mathrm{is\ even}}c_i(G)\right|.$$

Call $G$ balanced when $D_{clique}(G)\leq 1$.

Balanced-graph conjecture. Any flat graph is a balanced graph.

The source notes that related subclasses of flat graphs satisfy $D_{clique}(G)=1$. The supplied text gives no resolution of the conjecture, so it remains open.