The topological exactness conjecture for hard telic problems

Let $(X,T)$ be an efficiently discretizable topological dynamical system. Recall that $T$ is topologically exact if, for every nonempty open set $U\subset X$, there exists $n_0\in\mathbb{N}$ such that $T^{n_0}(U)=X$. A telic problem is called hard-on-average when it satisfies the stated almost-everywhere average-case hardness condition.

Topological exactness conjecture. If $T$ is topologically exact and the system admits a hard telic problem, then there exists an almost-everywhere $\delta$ hard-on-average telic problem coming from the dynamical system.

The conjecture is motivated by the possible construction of one-way functions from dynamical systems with hard telic problems. Its resolution is not supplied in the source.