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

Suppose that $M$ is a closed manifold of dimension greater than two and $r\ge 2$. We show that there exists a $C^r$-diffeomorphism $f:M⟶M$ with a wild affine blender-horseshoe ${\mathrm{\Lambda }}_f$ which is $C^r$-robustly and strongly pluripotent for ${\mathrm{\Lambda }}_f^{(\mathrm{m}\mathrm{j})}$ but not for ${\mathrm{\Lambda }}_f$, where ${\mathrm{\Lambda }}_f^{(\mathrm{m}\mathrm{j})}$ is the subset of ${\mathrm{\Lambda }}_f$ 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 ${\mathrm{\Lambda }}_f^{(\mathrm{m}\mathrm{j})}$ to the whole horseshoe.