Packing-minima transference conjecture

Let $K$ be a convex body in $\mathbb{R}^n$, let $K_c=\frac12(K-K)$ be its central symmetral, and let $\Lambda$ be a lattice in $\mathbb{R}^n$. Write $[K_c]^\star$ and $[\Lambda]^\star$ for the polar body and dual lattice, respectively, and let $\rho_i(K,\Lambda)$ denote the $i$th packing minimum.

Packing-minima transference conjecture. For $1\leq j\leq n$,

$$\rho_j(K,\Lambda)\,\rho_{n-j+1}([K_c]^\star,[\Lambda]^\star)\geq\frac12.$$

This is proposed as a complete packing-minima analogue of the lower bound in Banaszczyk's transference inequality. The statement is open in the source.