Optimal parameter-dependence conjecture for Brunn–Minkowski convex-hull defects

Let $n\in\mathbb{N}$ and $t\in(0,1/2]$, and let $A,B\subset\mathbb{R}^n$ be measurable sets of equal volume satisfying

$$|tA+(1-t)B|=(1+\delta)|A|,$$

where $\delta\in[0,d_{n,t}]$. Optimal parameter-dependence conjecture. There are computable constants $c_n,d_{n,t}>0$ such that

$$|\operatorname{co}(A)\setminus A|\leq c_nt^{-1}\delta|A|,$$

และ

$$|\operatorname{co}(B)\setminus B|\leq c_nt^{-n+1}\delta|A|.$$

The stated powers of $\delta$ and $t$ are motivated as optimal by examples in the source. The conjecture is established in dimension two, but remains unresolved in the generality stated.