On the degenerate Arnold conjecture on $\mathbb T^{2m}\times \mathbb C\mathbb P^n$

In the 1960s Arnold conjectured that a Hamiltonian diffeomorphism of a closed connected symplectic manifold $(M,\omega)$ should have at least as many contractible fixed points as a smooth function on $M$ has critical points. Such a conjecture can be seen as a natural generalization of Poincar\'e's last geometric theorem and is one of the most famous (and still nowadays open in its full generality) problems in symplectic geometry. In this paper, we build on a recent approach of the authors and Izydorek to the Arnold conjecture on $\mathbb C\mathbb P^n$ to show that the (degenerate) Arnold conjecture holds for Hamiltonian diffeomorphisms $\phi$ of $\mathbb T^{2m}\times \mathbb C\mathbb P^n$, $m,n\geq 1$, which are $C^0$-close to the identity in the $\mathbb C \mathbb P^n$-direction, namely that any such $\phi$ has at least $\text{CL}(\mathbb T^{2m}\times \mathbb C\mathbb P^n)+1= 2m+n+1$ contractible fixed points.