= Solution
Vary only the <probability density function> of $X$, using $\eta_t=\eta(1+tb)$ with bounded $b$ and $E_\eta b(X)=0$. The nuisance <score function> is $b(X)$, giving the <statistical tangent set>
$$
\{b(X):\ b\text{ bounded and measurable},\ E_\eta b(X)=0\}.
$$
Its closed linear span is the <nuisance tangent space> of all centered $L^2(\eta)$ <functions> of $X$, by <density of bounded centered scores>.
The <conditional expectation> of the parametric <score function> given $X$ is zero:
$$
E\left[\frac{h_\theta(X)\varepsilon}{\sigma^2}\,\middle|\,X\right]=0.
$$
It is therefore orthogonal to this <nuisance tangent space>. Its <orthogonal projection> onto that space vanishes, so the <efficient score> is unchanged. By <independence> and $E\varepsilon^2=\sigma^2$,
$$
\boxed{\widetilde\ell_{\theta,\eta}=\frac{h_\theta(X)\varepsilon}{\sigma^2},\qquad
\widetilde I_{\theta,\eta}=\frac{E_\eta h_\theta(X)^2}{\sigma^2}.}
$$
These equal the parametric <score function> and <Fisher information> when $\eta$ is known. \b[There is no loss of information from the unknown covariate density.] This is <adaptivity to an unknown covariate distribution>; it follows from score orthogonality, without having to estimate the <nuisance parameter>.
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