Conjugacy-class parity conjecture for finite 2-groups

Let $\pi$ be a finite $2$-group and let $N\triangleleft T\trianglelefteq W\triangleleft\pi$ be normal subgroups such that $\pi/N$ is cyclic, $T/N\cong\mathbb Z/{2}$, and $\pi/W\cong\mathbb Z/{2}$. Let $i\colon W\to\pi$ be inclusion and $p\colon\pi\to\pi/N$ the quotient map. Assume that

$$p_*\colon H_4(\pi;\mathbb Z/{2})\to H_4(\pi/N;\mathbb Z/{2})\cong\mathbb Z/{2}$$

and

$$(p\circ i)_*\colon H_2(W;\mathbb Z/{2})\to H_2(\pi/N;\mathbb Z/{2})\cong\mathbb Z/{2}$$

are nonzero, while

$$(p\circ i)_*\colon H_4(W;\mathbb Z/{2})\to H_4(\pi/N;\mathbb Z/{2})\cong\mathbb Z/{2}$$

is zero. Conjugacy-class parity conjecture. The number of $\pi$-conjugacy classes of elements in $T-N$ is odd. The statement is presented as a conjecture motivated by the study of vanishing of $\kappa'_4(\pi)$; no resolution is supplied in the provided text.