The h-vector and g-vector relation conjecture

Let $t_0\xrightarrow{k}t_1$ be adjacent vertices with $B^1=\mu_k(B^0)$, and let $h_k,h'_k$ and $g_k$ be the integers defined from the relevant $F$-polynomial specializations and the $\mathbf{g}$-vector. h-vector and g-vector relation conjecture. In the notation of Proposition 3.7 and the definition of $h_k$,

$$h'_k=-[g_k]_+,\qquad h_k=-[-g_k]_+=\min(0,g_k).$$

This sharpens the proved relation $g_k=h_k-h'_k$. The conjecture is established for the bipartite belt but is open in general.