On the same $N$-type conjecture for the suspension of the infinite complex projective space

Let $[\varphi _{i_{k}},[\varphi _{i_{k-1}},\cdots,[\varphi _{i_{1}}, \varphi _{i_{2}}],\cdots ]]$ be an iterated commutator of self-maps $\varphi _{i_{j}}$ on the suspension of the infinite complex projective space. In this paper, we produce useful self-maps of the form $I + [\varphi _{i_{k}},[\varphi _{i_{k-1}},\cdots, [\varphi _{i_{1}}, \varphi _{i_{2}}],\cdots ]]$, where $+$ means the addition of maps on the suspension structure of $\Sigma {\mathbb{C}}P^{\infty}$. We then give the answer to the conjecture saying that the set of all the same homotopy $n$-types of the suspension of the infinite complex projective space is the one element set consisting of a single homotopy type.