The contraction conjecture for dual volumes of ball bodies

Let $n\ge2$, let $T:\mathbb R^n\to\mathbb R^n$ be a contraction, and let $K\in\mathcal S_n$.

Contraction dual-volume conjecture. Then

$${\rm Vol}((TK)^c)\le {\rm Vol}(K^c).$$

This is presented as a variant of the Kneser–Poulsen intersection conjecture in the language of $c$-duality. Its resolution is not specified in the source.