= Solution
The positive-frequency solution for $u_k=af_k$ is proportional to $e^{-ick\tau}$. Since $a=-1/(H\tau)$, its scalar-field mode function has the stated form $f_k=\mathcal N\tau e^{-ick\tau}$. The <canonical momentum> in conformal time is $\pi=a^2\varphi'$, so the <canonical commutation relation> requires the <Wronskian normalization>
$$
a^2(f_kf_k^{*\prime}-f_k^*f_k')=i.
$$
Substitution gives $2ick\,a^2|\mathcal N|^2\tau^2=i$. Hence, up to an irrelevant constant phase,
$$
\boxed{\mathcal N=\frac{H}{\sqrt{2ck}}},
\qquad
f_k(\tau)=\frac{H\tau}{\sqrt{2ck}}e^{-ick\tau}.
$$
The choice $e^{-ick\tau}$ is the <Bunch-Davies vacuum> condition at early conformal time.
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