Kleitman's conjecture on maximal intersecting subfamilies of the Boolean lattice

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote its power set, and let $\mathcal{F}$ be a maximal intersecting subfamily of $2^{[n]}$. For each $A\subseteq[n]$, let $e_A$ be the corresponding standard basis vector of $\mathbb{R}^{2^{[n]}}$. Define the embedding

$$\vec{\mathcal{F}}=\sum_{A\subseteq[n]}e_A\begin{cases}+1,&\text{if }A\in\mathcal{F}\text{ and }A^c\notin\mathcal{F},\\-1,&\text{if }A\notin\mathcal{F}\text{ and }A^c\in\mathcal{F},\\0,&\text{otherwise.}\end{cases}$$

For $a\in[n]$, let $\mathcal{S}_a(2^{[n]})=\{A\subseteq[n]:a\in A\}$ and let $\vec{\mathcal{S}}_a(2^{[n]})$ denote its embedding. Kleitman's conjecture. There exist non-negative numbers $c_a$ and $\lambda_{(A,B)}$ such that

$$\vec{\mathcal{F}}=\sum_{a\in[n]}c_a\vec{\mathcal{S}}_a(2^{[n]})+\sum_{A\subseteq B\subseteq[n]}\lambda_{(A,B)}(e_B-e_A),$$

where $\sum_{a\in[n]}c_a=1$.

Kleitman proposed this as a strengthening of Chvátal's conjecture by characterizing the embedded vectors of maximal intersecting subfamilies of the Boolean lattice. The paper states partial results and constructions but does not report a resolution, so the conjecture is recorded as open.