Let $P_\bullet$ be a non-negative chain complex in an abelian category $\mathcal A$ such that $H_n(P_\bullet) \cong 0$ and each $P_n$ is projective. Then each $Z_{n} = \partial(P_{n+1})$ is projective.
This is for Proposition 2.5.1.10 at Kerodon. It starts using $H_n(P_\bullet) = B_n/Z_n \cong 0$ to get a short exact sequence $$ 0\to Z_n \to P_n\xrightarrow\partial Z_{n-1}\to 0. $$
In particular $Z_0\cong P_0$ is projective.
Now suppose that $Z_{n-1}$ is projective Splitting the exact sequence we get $$ P_n \cong Z_{n}\oplus Z_{n-1}. $$ We're done if $\mathcal A$ is category of chain complexes of modules for then $$ P \text{ projective} \iff P\oplus Q \text{ free for some } Q $$
So $Z_n\oplus Z_{n-1}\oplus Q $ is free and $Z_n$ is projective.
However, I don't know how to proceed in general abelian categories.