The triviality of fibers conjecture for unicritical and exponential families

Consider the family of unicritical polynomials of degree $d\geq2$ or the exponential family. A parameter fiber is trivial when it contains only the parameter itself; fibers are defined using separation lines formed from hyperbolic, parabolic, periodic, and preperiodic parameter-ray data. Triviality of fibers conjecture. All fibers are trivial. Trivial fibers imply local connectivity at the corresponding parameters for Multibrot sets, so the conjecture is a parameter-space analogue of local connectivity. The survey does not report a resolution in the stated generality.