The differentiable-map conjecture on closed special alpha-limit sets

Let $f:[0,1]\to[0,1]$ be continuously differentiable, and let $s\alpha(x)$ denote a special $\alpha$-limit set. Differentiable-map conjecture. Every special $\alpha$-limit set of $f$ is closed. The paper states this conjecture as unresolved; it is a proposed sufficient condition for all special $\alpha$-limit sets to be closed.