As requested in the comments, I'm writing an answer with references and a short explanation. Because I don't deeply understand some step, I hope someone in the future will clarify this post.

You can find it on page 5-6 (293-294) of Taylor's "On triple Siegel l-adic cohomology" or on page 10 of Mokrane and Tilouine "Cohomology of Siegel manifolds with p-adic integral coefficients and applications".

The proof in the second reference I quote above is a bit clearer (to me, at least).

First of all, it is a known fact that it remains to show that $H^{k}_{\text{cusp}}=H^{k}_{(2)}$ where the right hand side denotes the $L^{ 2}$-cohomology. Now, according to Matsushima's formula, it remains to show that any representation appearing in $H_{(2)}$ is cupsidal and appears with the same multiplicity as in $H_{\text{cusp}}$. The Vogan-Zuckerman classification tells you that if $\pi_{\infty}$ appears in $H_{(2)}^{3}$ then it is tempered (this is why being with $\mathbf{GSp}_{4}$ is important but it is unclear for me, see bellow). Finally, a theorem of Wallach claims that for such $\pi_{\infty}$ the representations $\pi_{\infty}\pi_{f}$ are cuspidal and $\pi_{\infty}$ occur with the multiplicity desired.

I will simply conclude with a few words on the Vogan-Zuckerman classification. The paper quoted is D. Vogan, G. Zuckermann "Unit representations with non-zero cohomology". I'm having trouble finding where in this article is a result that allows us to prove that any $L^{2}$-cohomological representation of $\mathbf{GSp}_{4}$ is tempered. Maybe someone can comment or add an answer.

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