Suppose $X$ is a pre-simplicial set defined on a finite vertex set $\{v_1, v_2, \ldots, v_k\}$ recursively as follows:

Let $X_1$ have a single vertex $\{v_1\}$.

For $i > 1$, $X_i$ is obtained from $X_{i-1}$ by attaching the cone of a subcomplex $S \subseteq X_{i-1}$, $X_i = X_{i-1} \cup CS$, where we label the apex of $CS$: $v_i$.

$X = X_k$.

The point of the construction is that $X$ is built by successively including more simplices, and never identifying any of them. My question is this: Is $H_*(X)$ torsion-free? Small examples seem to indicate 'yes.' A cursory search of the internet does not turn up any results.

Note, if a simplicial complex $Y$ is merely torsion-free, then $Y \cup CS$ need not be torsion-free, so the recursive construction of $X$ has to play a major role in preventing torsion (if the answer to the question turns out to be yes).

Any references or ideas would be greatly appreciated!