Abrar's dominant dimension bound for Nakayama algebras

Let $A$ be a non-self-injective Nakayama algebra with $n\geq 3$ simple modules, and let $\operatorname{domdim}(A)$ denote its dominant dimension.

Abrar's conjecture.

$$\operatorname{domdim}(A)\leq 2n-3.$$

The source later states that this conjectural bound is corrected and proved false: the dominant dimension is instead bounded by $2n-2$. Thus the displayed conjecture is refuted.