The coincidence, density, and dynamical-relation conjecture for marked points

Let $V$ be an irreducible quasiprojective complex algebraic variety, and let $f:V\times\mathbb{P}^1\to\mathbb{P}^1$ be an algebraic family of rational maps of degree $d\geq 2$, with dimension $N>0$ in moduli. For marked points $a_0,\ldots,a_N$, let $S(a_0,\ldots,a_N)$ be the set of parameters where every $a_i(t)$ is preperiodic for $f_t$. The points are coincident along $V$ if, on a Zariski-open subset, preperiodicity of all but one point implies preperiodicity of the remaining point; they are dynamically related along $V$ when they satisfy an $f$-invariant algebraic relation. The coincidence, density, and dynamical-relation conjecture. For any collection of $N+1$ marked points $a_0,\ldots,a_N$, the following are equivalent: (1) $S(a_0,\ldots,a_N)$ is Zariski dense in $V$; (2) the points $a_0,\ldots,a_N$ are coincident along $V$; and (3) the points $a_0,\ldots,a_N$ are dynamically related along $V$. This connects simultaneous preperiodicity with algebraic and dynamical dependence in families of rational maps; the supplied text proves implications involving dynamical relations and coincidence but does not establish a resolution of the full equivalence.