Call–Silverman conjecture on variation of dynamical degrees

Let $T$ be an irreducible variety, let $X/T$ be a family of smooth irreducible projective varieties, and let $f:X/T\dashrightarrow X/T$ be a family of dominant rational maps. Write $f_\eta$ for the map on the generic fiber and $f_t$ for the map on the fiber over $t\in T$, with dynamical degrees $\delta(f_\eta)$ and $\delta(f_t)$. Call–Silverman conjecture. For every $\epsilon>0$, the set

$$\bigl\{t\in T:\delta(f_t)\leq\delta(f_\eta)-\epsilon\bigr\}$$

is not Zariski dense in $T$. This asserts that dynamical degree does not drop by a fixed positive amount on a Zariski-dense set of fibers. The paper states that Xie proved the projective-space reduction case, while the broader family formulation remains a natural generalization.