The duality conjecture for covering numbers

Let $K$ and $L$ be symmetric convex bodies in $\mathbb R^n$, and let $N(K,L)$ be the covering number: the least number of translates of $L$ needed to cover $K$. For a symmetric convex body $K$, write $K^*$ for its polar body.

Covering-number duality conjecture. There are universal constants $c,C>0$ such that, for every dimension $n$ and every pair of symmetric convex bodies $K,L\subset\mathbb R^n$,

$$N(K,L)\leq N(L^*,cK^*)^C.$$

This is presented as an open problem in geometric analysis. It proposes a dimension-free comparison between a covering number and the covering number of the corresponding polar bodies.