Nevo–Petersen–Tenner balanced-complex conjecture for gamma vectors of flag spheres

Let $K$ be a flag simplicial sphere, and let $\gamma(K)$ denote its gamma vector. A simplicial complex is balanced if, for its vertex set $V$, there is a map $\kappa:V\to[d]$ such that adjacent vertices receive distinct colors; equivalently, every face has vertices of distinct colors. Nevo–Petersen–Tenner conjecture. The gamma vector $\gamma(K)$ is the $f$-vector of some balanced simplicial complex. This conjecture proposes a combinatorial realization of the gamma vector, strengthening its mere entrywise nonnegativity; its resolution status is not specified in the source.