= Solution
Under the null model, $\pi=1$, which is a boundary point of the parameter space. Standard interior-point asymptotic normality of the <maximum-likelihood estimator> therefore need not apply.
Indeed, put $a=1-e^{-\lambda}$. Under $\pi=1$, $N_+\sim\operatorname{Binomial}(n,a)$. For the unconstrained estimator $\widetilde\pi=N_+/(na)$, the <central limit theorem> gives
$$
\sqrt n(\widetilde\pi-1)
\xrightarrow{d}W,
\qquad
W\sim N\!\left(0,\frac{1-a}{a}\right)
=N\!\left(0,\frac{e^{-\lambda}}{1-e^{-\lambda}}\right).
$$
Since $\widehat\pi=\min\{1,\widetilde\pi\}$, the <continuous mapping theorem> yields
$$
\boxed{Z=\min\{0,W\}.}
$$
This distribution has probability $1/2$ at zero and a continuous normal half-density on the negative half-line, so it is not normal.
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