Nandakumar–Ramana Rao conjecture for convex bodies

Let $K$ be a planar convex body and let $n>1$ be a natural number. A partition of the plane into $n$ convex pieces is a collection of convex pieces $P_1,\ldots,P_n$ whose union is the plane. Nandakumar–Ramana Rao conjecture. There exists such a partition satisfying

$$\mathrm{area}(P_1\cap K)=\cdots=\mathrm{area}(P_n\cap K)$$

and

$$\mathrm{perimeter}(P_1\cap K)=\cdots=\mathrm{perimeter}(P_n\cap K).$$

The case $n=2$ was proved by Nandakumar and Ramana Rao, and the case $n=3$ was later resolved by Bárány, Blagojević, and Szűcs using topological methods; the general case remains open.