The generalized joints bound for flats

Let $d_1,\dots,d_r$ and $m_1,\dots,m_r$ be positive integers, put $d=d_1m_1+\cdots+d_rm_r$, and let a $(d_1^{m_1},\dots,d_r^{m_r})$-joints configuration $(\mathcal J,\mathcal F_1,\dots,\mathcal F_k)$ consist of a family $\mathcal F_i$ of $d_i$-flats in $\mathbb F^d$ for each $i\in[r]$, together with joints formed by the intersection of $m_i$ members of each family, not all lying in a hyperplane. Generalized joints conjecture. For every $\bm d=(d_1^{m_1},\dots,d_r^{m_r})$, there is a constant $C_{\bm d}$ such that every such configuration satisfies

$$|\mathcal J|\leq C_{\bm d}\bigl(|\mathcal F_1|^{m_1}\cdots|\mathcal F_r|^{m_r}\bigr)^{1/(m_1+\cdots+m_r-1)}.$$

This would extend the joints and multijoints bounds to mixtures of flats of different dimensions. Over $\mathbb R$, the source cites an existing bound with an $o(1)$ loss, while the stated uniform bound is left open.