The relative critical height conjecture for regular polynomial endomorphisms

Let $\mathsf{M}_d^N$ be the moduli space of endomorphisms of $\mathbb{P}^N$, let $\mathsf{P}_d^N\subseteq\mathsf{M}_d^N$ be the moduli space of regular polynomial endomorphisms, and let restriction to the invariant hyperplane define $\pi:\mathsf{P}_d^N\to\mathsf{M}_d^{N-1}$. For an endomorphism $f$ with invariant hyperplane $H$, define the relative critical height by

$$\hat{h}_{\mathrm{relcrit}}(f):=\hat{h}_{\mathrm{crit}}(f)-\hat{h}_{\mathrm{crit}}(f|_H)=\hat{h}_{\mathrm{crit}}(f)-\hat{h}_{\mathrm{crit}}(\pi(f)).$$

For any ample Weil heights $h_{\mathsf{P}_d^N}$ and $h_{\mathsf{M}_d^{N-1}}$ on $\mathsf{P}_d^N$ and $\mathsf{M}_d^{N-1}$, respectively, the relative critical height is a relative moduli height. For every such $f$,

$$\hat{h}_{\mathrm{relcrit}}(f)\asymp h_{\mathsf{P}_d^N}(f)+O\bigl(h_{\mathsf{M}_d^{N-1}}(f|_H)\bigr).$$

The relative critical height is non-negative, compatible with iteration, and vanishes precisely on maps whose critical orbits are no more complex than required by their behaviour at infinity. A sufficiently precise form, together with the corresponding conjecture in dimension one lower, would imply Silverman's conjecture for regular polynomial endomorphisms; the proposed relative-height statement itself remains open.