Equality characterization for independent distance domination in connected bipartite graphs

Let $d\geq 2$, and let $G$ be a connected bipartite graph of order $n$. Write $C_{2d+2}$ for the cycle of length $2d+2$, and let $\mathcal{B}_d$ denote the graph family defined in the source. The invariant $\gamma_d^1(G)$ is the minimum size of a $d$-distance dominating independent set in $G$. The bipartite equality conjecture.

$$\gamma_d^1(G)=\frac{n}{d+1}\quad\Longleftrightarrow\quad G\in\{C_{2d+2}\}\cup\mathcal{B}_d\ \text{ or }\ n=d+1.$$

This conjecture seeks a complete characterization of equality in the upper bound for connected bipartite graphs. The supplied source recalls the corresponding characterization for $d=1$, while the asserted case $d\geq 2$ is posed as open.