Triviality of fibers for quadratic polynomials and exponential maps

Consider the parameter spaces of the quadratic family $p_c(z)=z^2+c$ and the exponential family $E_c(z)=e^z+c$. For a parameter $c$, its reduced fiber is the set of parameters in the reduced parameter space that cannot be separated from $c$ by any separation line; a fiber is trivial when it consists of exactly one point.

Trivial-fibers conjecture. In the parameter spaces of quadratic polynomials and exponential maps, all fibers are trivial.

For quadratic polynomials, this conjecture is equivalent to the local connectivity conjecture for the Mandelbrot set. The exponential case requires substantially subtler topological information about parameter space and remains open.