A subset \(S\) of the vertex set of a graph \(G\) is called acyclic if the subgraph it induces in \(G\) contains no cycles. We call \(S\) an acyclic dominating set if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by \(\gamma_a(G)\), is called the acyclic domination number of \(G\). A graph \(G\) is \({2-diameter-critical}\) if it has diameter \(2\) and the deletion of any edge increases its diameter. In this paper, we show that for any positive integers \(k\) and \(d \geq 3\), there is a \(2\)-diameter-critical graph \(G\) such that \(\delta(G) = d\) and \(\gamma_a(G) – \delta(G) \geq k\), and our result answers a question posed by Cheng et al. in negative.