Generalized Nandakumar–Ramana Rao conjecture for convex bodies and measures

Let $K$ be a convex body in $\mathbb{R}^d$, let $\mu$ be an absolutely continuous probability measure on $\mathbb{R}^d$, let $n>1$ be a natural number, and let $\varphi_1,\ldots,\varphi_{n-1}\colon\operatorname{Conv}(\mathbb{R}^d)\to\mathbb{R}$ be continuous functions. A partition of $\mathbb{R}^d$ into $n$ convex pieces is a collection of convex pieces $P_1,\ldots,P_n$ whose union is $\mathbb{R}^d$. Generalized Nandakumar–Ramana Rao conjecture. There exists such a partition satisfying

$$\mu(P_1\cap K)=\cdots=\mu(P_n\cap K)$$

and, for every $i\in\{1,\ldots,n-1\}$,

$$\varphi_i(P_1\cap K)=\cdots=\varphi_i(P_n\cap K).$$

This extends the planar area-and-perimeter problem to arbitrary dimension, absolutely continuous measures, and arbitrary continuous functionals; the source gives the planar cases $n=2$ and $n=3$ as resolved, while the generalized assertion is not resolved there.