Orbit propagation principles for rational points

Let $K$) be a number field, let $X/K$ be a smooth projective variety, and let $f:X\to X$ be an endomorphism defined over $K$. For $P\in X$, write $\mathcal{O}_f(P)=\{f^n(P):n\geq 0\}$, let $\mathcal{O}_f^{\textup{grand}}(P)=\{Q\in X:\mathcal{O}_f(P)\cap\mathcal{O}_f(Q)\ne\emptyset\}$, write $P\mathbin{\equiv_f}Q$ when their forward orbits meet, and set $X_f^{\textup{dense}}=\{P\in X:\overline{\mathcal{O}_f(P)}=X\}$. Assume that $X(K)$ has at least one Zariski dense $f$-orbit. Then there is a finite extension $K'/K$ such that all of the following hold: Orbit propagation principles. For all $P_1,\ldots,P_r\in X(K')$, each of the complements of their union of forward orbits and of their union of grand orbits is Zariski dense in $X$; $X(K')$ contains a Zariski dense set of representatives for $X(K')/\mathbin{\equiv_f}$; and every set of representatives in $X(K')$ for $X_f^{\textup{dense}}(K')/\mathbin{\equiv_f}$ is Zariski dense in $X$. These principles formalize the claim that one dense orbit propagates to many widely distributed rational orbits; the source presents them as conjectural propagation statements, and no resolution is supplied here.