The continuous face-vector representation conjecture for flag complexes

Let $(1,c_1,c_2,\dots,c_d)$ be the face vector of a flag complex. Consider nonnegative real numbers $a_j^k$ for all integers $1\leq j\leq k\leq d$.

Continuous face-vector representation conjecture. There are such numbers satisfying, for every $1\leq j\leq d$,

$$c_j=\sum_{i_1<i_2<\dots<i_j}a_{i_1}^{i_j}a_{i_2}^{i_j}\dots a_{i_j}^{i_j},$$

with $a_i^k\geq a_j^k$ whenever $i<j$, and $a_j^i\geq a_j^k$ whenever $i<k$.

The conjecture gives a continuous analogue of the construction of flag complexes from complete multipartite graphs and would provide a characterization of flag-complex face vectors up to the integrality issue. The source presents it as an open conjecture motivating the subsequent construction and approximation arguments.