To provide another answer (or rather a hint, as it is an exercise): You can indeed, as Richard suggested, use the characterization theorem using multicomplexes. More elementary, you can use the following idea: the contribution to h_i appears when along a shelling, the minimal new face is of cardinality face. Consider the complementary face, of cardinality d-i (if the complex is of dimension d-1). This is the minimal "old" face.

Now, there are at least two contributions to h_i. You can assume that the minimal old faces are disjoint (by passing to the link of a intersection). Similarly, you can pass to links and assume that the minimal new face is disjoint. You can now apply your argument to each face separately.

To provide another answer (or rather a hint, as it is an exercise): You can indeed, as Richard suggested, use the characterization theorem using multicomplexes. More elementary, but a little less insightful, you can use the following idea: the contribution to $h_i$ appears when along a shelling, the minimal new face is of cardinality face. Consider the complementary face, of cardinality $d-i$ (if the complex is of dimension $d-1$). This is the minimal "old" face. Note that if the minimal old faces do not coincide, then by your argument, you immediately get two generators of $h_i-1$ in the respective stars of old faces.

But if they do have an intersection, you can pass to the link of the intersection face. Hence, you end up in the case where $i=$ dimension of the complex $+1$.

Now, if the minimal new faces (the entire cardinality $i$-face, as we know now) of both generators intersect, you can once again pass to a link, unless $i=2$. You end up in one of two cases: Either the two generators are disjoint, or $i=2$ and you are in a graph. Either case is a somewhat easy exercise.