Wedge-of-spheres conjecture for robust clique complexes of square-sequence graphs

Let $G$ be a graph with a square sequence

$$C_4 \cong H_1 \subset H_2 \subset \cdots \subset H_n \cong G,$$

where $n\geq 1$, and let $\mathsf{Cliq}_{k}(G)$ denote the $k$-robust clique complex of $G$.

Wedge-of-spheres conjecture. For all $k\geq 2$, the complex $\mathsf{Cliq}_{k}(G)$ is homotopy equivalent to a wedge sum of $(2k-3)$-dimensional spheres. Moreover, the number of spheres is determined by a recurrence relation based on the homotopy type of $\mathsf{Cliq}_{k}(H_{n-1})$ and the specific attachment of the $n$-th square.

This conjecture seeks to extend the established homotopy results for robust clique complexes from $k\in\{2,3\}$ to all $k\geq 2$. The stated recurrence is not given explicitly here, so determining the sphere count and proving the claimed wedge decomposition remain open.