Full-dimensional parameter set for the CS property

Let $U\subset\mathbb R^k$ be the parameter space of the iterated function systems $\Phi_u$, and let $\mathfrak m$ be a slowly decaying $\sigma$-invariant ergodic probability measure satisfying

$$\mathfrak h(\mathfrak m)>-\bigl(\lambda_1(\mathfrak m,u)+\cdots+\lambda_d(\mathfrak m,u)\bigr)$$

for Lebesgue almost every $u\in U$. Full-dimensional CS-property conjecture. Under some weak assumptions on the maps $A_i$ and $t_i$, there exists $U'\subset U$ such that $\dim_H(U')=k$, and for every $u\in U'$, the IFS $\Phi_u$ satisfies the CS property with respect to $\mathfrak m$. The CS property is the overlap-related property defined in the paper; the supplied text gives no resolution and only motivates the conjecture by analogy.