Complex Gr"unbaum lower-bound conjecture for regular fans

Let $m$ be a positive integer, let $k\geq 1$, and let $q_1,\ldots,q_k>2$ be integers. Set

$$Q=\prod_{j=1}^k q_j.$$

Here $\Delta_{\mathbb{C}}(m;q_1,\ldots,q_k)$ is the minimum dimension $d$ such that any $m$ measures on $\mathbb{C}^d$ can be equipartitioned by $Q$ regions determined by $k$ complex regular $q_j$-fans. Complex Gr"unbaum lower-bound conjecture.

$$k\Delta_{\mathbb{C}}(m;q_1,\ldots,q_k)\geq m\left\lfloor\frac{Q-1}{2}\right\rfloor.$$

This conjecture extends the known lower bound for a single complex regular fan to arbitrary numbers of fans and is motivated by the moment-curve argument used in the real Gr"unbaum problem. Its status is not resolved in the supplied source.