The optimal rearrangement-constant conjecture for BMO

Let $Q$ be a $d$-dimensional cube and let $f^*$ denote the monotone rearrangement of $f$ on $[0,1]$. Let $c(d)$ be the constant appearing in the dimension-reduction conjecture for the John–Nirenberg Bellman function. The optimal rearrangement-constant conjecture. The constant $c(d)$ is the best possible constant in

$$\|f^*\|_{\mathrm{BMO}([0,1])}\leqslant c(d)\|f\|_{\mathrm{BMO}(Q)}.$$

This connects the Bellman-function dimension-reduction problem with monotone rearrangement estimates. The source states the optimality claim as a conjecture and supplies no resolution.