A robust obstruction to full strong pluripotency for wild blender-horseshoes

Suppose that $M$ is a closed manifold of dimension greater than two and $r\geq 2$. We show that there exists a $C^r$-diffeomorphism $f:M\longrightarrow M$ with a wild affine blender-horseshoe $\Lambda_f$ which is $C^r$-robustly and strongly pluripotent for $\Lambda_f^{(\mathrm{mj})}$ but not for $\Lambda_f$, where $\Lambda_f^{(\mathrm{mj})}$ is the subset of $\Lambda_f$ consisting of elements with the majority condition. Thus, within the present affine blender-horseshoe family in [KNS], there is a robust obstruction to extending the strong pluripotency from $\Lambda_f^{(\mathrm{mj})}$ to the whole horseshoe.