Brannen's simplex conjecture for projection-body volume

Theorem 1: for $K=T_{10}\times T_{10}\subset\mathbb R^{20}$, $R_{20}(K)/c_{20}=121\binom{20}{10}/(21\cdot2^{20})>1$, so the simplex bound fails in dimension 20; iterating the product identity $R_{r+s}(A\times B)=R_r(A)R_s(B)$ gives $R_n(K_n)\ge\lambda^nc_n$ for some $\lambda>1$ and all large $n$. Priority: Feng, Hu, Liu and Xu (preprint, August 2026) already disproved the conjecture in every dimension $d\ge9$, and Chen et al. proved it in dimension 3; the paper cites both and presents its result as a direct product construction with an exact value. It does not determine the optimal upper constant or the maximizers.

Theorem 1: for $K=T_{10}\times T_{10}\subset\mathbb R^{20}$, $R_{20}(K)/c_{20}=121\binom{20}{10}/(21\cdot2^{20})>1$, so the simplex bound fails in dimension 20; iterating the product identity $R_{r+s}(A\times B)=R_r(A)R_s(B)$ gives $R_n(K_n)\ge\lambda^nc_n$ for some $\lambda>1$ and all large $n$. Priority: Feng, Hu, Liu and Xu (preprint, August 2026) already disproved the conjecture in every dimension $d\ge9$, and Chen et al. proved it in dimension 3; the paper cites both and presents its result as a direct product construction with an exact value. It does not determine the optimal upper constant or the maximizers.