Thickness bounds imply recursively bounded refinement complexity

Let $K^n$ be a simplicial complex, let $\epsilon$ be positive, and let $B^d_1\subset\mathbb{R}^d$ denote the unit ball. A piecewise smooth embedding of $K^n$ into $\mathbb{R}^d$ has thickness $\epsilon$ when it satisfies the paper's thickness condition. Thickness–refinement conjecture. If $K^n$ admits a piecewise smooth embedding of thickness $\epsilon$ into $\mathbb{R}^d$, more specifically into $B^d_1\subset\mathbb{R}^d$, then

$$\operatorname{rc}(K)<r(\epsilon),$$

where $r$ is a recursively computable function of $\epsilon$. This technical conjecture would connect geometric thickness with refinement complexity and is intended to support a thickness analogue of the non-recursive refinement-complexity theorem. The paper notes that proving it is technically difficult because its minimal thickness definition does not control the proximity of adjacent simplices.