The scaled face-vector conjecture for flag complexes

Let $(1,c_1,c_2,\dots,c_d)$ be a vector of positive integers, and let $a_j^k$ be real numbers for all integers $1\leq j\leq k\leq d$ satisfying

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

for every $1\leq j\leq d$, together with $a_i^k\geq a_j^k$ whenever $i<j$ and $a_j^i>a_j^k$ whenever $i<k$.

Scaled face-vector conjecture. There is an integer $q$ such that $(1,qc_1,q^2c_2,\dots,q^dc_d)$ is the face vector of a flag complex.

This is intended to overcome the nonintegrality of the continuous multipartite construction: after a common scaling, the prescribed real parameters should yield an actual flag complex. The source presents the statement as part of its proposed route toward characterizing flag-complex face vectors.