The extended multiplier-index conjecture for polynomial maps

Let $V_d$ be the parameter space of multiplier data for degree-$d$ polynomial maps, let $\bar\lambda$ denote the corresponding parameter value, and let $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$ be the combinatorial data defined in the main theorem. The main theorem states that the converse of its emptiness criterion holds for $d\leq 7$, and that inclusion of these data gives a weak inequality between the corresponding fibers.

Extended multiplier-index conjecture. The converse of the emptiness criterion also holds when $d\geq 8$. Moreover, if

$$\mathcal{I}(\lambda)\subsetneq\mathcal{I}(\lambda')\quad\text{and}\quad\mathcal{K}(\lambda)\subseteq\mathcal{K}(\lambda'),$$

for $\lambda,\lambda'\in V_d$, then

$$\#\left(\Phi_d^{-1}\left(\bar\lambda\right)\right)>\#\left(\Phi_d^{-1}\left(\bar\lambda'\right)\right).$$

These assertions strengthen the corresponding parts of the main theorem: the first extends the converse from degrees at most $7$ to degrees at least $8$, while the second strengthens the fiber-cardinality inequality to a strict inequality when the inclusion for $\mathcal{I}$ is strict. The supplied text gives no resolution status.