Dombi's obstruction conjecture for strict monotonicity of higher-fold representation functions

Let $A\subseteq\mathbb N$ and let $h\geq 1$ be an integer. Define the ordered $h$-fold representation function by

$$R_{A,h}(n)=\#\{(a_1,\ldots,a_h)\in A^h:a_1+\cdots+a_h=n\}.$$

The complement $\mathbb N\setminus A$ is co-infinite when it is infinite. Dombi's conjecture. If $\mathbb N\setminus A$ is infinite, then $R_{A,h}(n)$ cannot be eventually strictly increasing. This conjecture proposes an obstruction to strict monotonicity for representation functions of sets with infinite complement; the provided text gives no resolution, while Dombi's earlier result shows that eventual non-strict increase can occur for suitable co-infinite sets when $h>4$.