Let $\Delta$ be an abstract simplicial complex which is pure (i.e. all facets have same cardinality) of dimension $d-1\ge 0$ and is shellable, with a shelling order of the facets given by $F_1,...,F_m$ i.e. for each $i>1$, the simplicial complex $\langle F_i\rangle \cap \langle F_1,...,F_{i-1}\rangle$ is generated by some faces of the form $F_i\setminus \{v\}$ for some $v\in F_i$. For each $i=2,...,m$, let $r_i$ be the no. of facets of $\langle F_i\rangle \cap \langle F_1,...,F_{i-1}\rangle$, and set $r_1=0$. Then it is known that the $i$-th component of the $h$-vector of $\Delta$ is given by $$h_i=|\{j: r_j=i\}|, \forall i=0,1,...,d.$$ (The $h_i$'s are such that $\dfrac{\sum_i h_it^i}{(1-t)^d}$ is the Hilbert-Series of the Stanley-Reisner ring of $\Delta$.)
My question:
If for some $i>1$, it holds that $r_j\ne i, \forall j=1,...,m$, then is it true that $r_j\ne i+1, \forall j=1,...,m$ ?