Connectivity conjecture for simplicial complexes of projective-cube walk powers

Let $r$ and $k$ be integers with $r\geq k$. Consider the simplicial complex associated to the graph $\mathcal{PC}_{2r}^{(2k-1)}$, where $\mathcal{PC}_{2r}^{(2k-1)}$ is the $(2k-1)$-st walk power of the projective cube $\mathcal{PC}_{2r}$. Connectivity conjecture. The associated simplicial complex is $2^{2k}$ connected. This is proposed as a strengthening of the walk-power chromatic-number conjecture, with algebraic-topological methods suggested as a possible approach; it remains open in the paper.