k-point suspension of spheres

Question

In "Complexes of Directed Trees" Kozlov defines the $k$-point suspension of a simplicial complex $X$ as $$susp_k(X) = \{k \text{ distinct points}\}*X,$$ were $*$ denotes the join of simplicial complexes.
As an easy example $susp_1(X) = Cone(X)$, and for $k=2$ this coincides with the ''usual'' suspension.
Is there a reference for the statement $$susp_k\left(\bigvee^m S^n\right) \sim \bigvee^{m(k-1)} S^{n+1}$$ when $k\ge2$ (here $\sim$ denotes homotopic equivalence)?

Answer 1

I found a result that should imply the statement above, in Lemma 2.4 of Directed Subgraph Complexes by Axel Hultman.

Answer 2

Not a reference but you can just start to construct this space and you will see why this is the case. For $k = 0$ and $k = 1$ this is immediate. Now let's increase $k$ by one $-$ add one more point. You get a picture like that ($m = 2, n=1$):

It is quite clear that you have added $m$ more spheres: contract the thick vertical segment on its lower end and contract the thick equators into south poles $-$ you will get $S^2 \vee S^2$ sitting inside a bigger $S^2 \vee S^2.$

You can imagine next points you add to be each above the previous one, with the corresponding segments together "embracing" the construction below. Then contracting vertical segments and equators will give you $k-1$ copies of $\bigvee_m S^{n+1}$ sitting inside each other.