The adjacency dimension bound for the join of a graph with a universal vertex

Let $H$ be a graph of order $n\ge 2$, let $K_1+H$ denote the graph obtained by adding a universal vertex to $H$, and let $\mathcal{C}(K_1+H)$ be its adjacency threshold. For $k\in\{1,\ldots,\mathcal{C}(K_1+H)\}$, write $\operatorname{adim}_k(G)$ for the $k$-adjacency dimension of a graph $G$. Adjacency dimension bound.

$$\operatorname{adim}_k(K_1+H)\le\operatorname{adim}_k(H)+k.$$

The bound gives an upper estimate for the adjacency dimension of a graph formed by adjoining a universal vertex, and the surrounding example shows that it can be tight. The supplied text does not state whether the bound has been proved or remains open.