= Solution
In the constant-$H$ <de Sitter approximation>, $a=-1/(H\tau)$ with $\tau<0$, so $a''/a=2/\tau^2$. The massive rescaled mode obeys
$$
v_{\mathbf k}''+\left[k^2+\frac{m_\chi^2/H^2-2}{\tau^2}\right]v_{\mathbf k}=0.
$$
For a fixed mass in the specified open interval, at sufficiently late <superhorizon scales> with $|k\tau|\ll1$, the $k^2$ term is negligible compared with the inverse-square term in the limiting equation. Try $v\propto(-\tau)^p$. The indicial equation is $p(p-1)+m_\chi^2/H^2-2=0$, giving
$$
\boxed{p=\frac12\pm\nu,\qquad
\nu=\sqrt{\frac94-\frac{m_\chi^2}{H^2}}.}
$$
Thus both independent branches of the <massive de Sitter superhorizon scalar mode> are present in
$$
\boxed{v_{\mathbf k}=\frac1{\sqrt{2k}}\left[
c_-(k)(-k\tau)^{1/2-\nu}+c_+(k)(-k\tau)^{1/2+\nu}\right].}
$$
The coefficients are constant in time and fixed by initial conditions. Writing $-k\tau>0$ avoids ambiguous fractional powers of the negative number $k\tau$; a chosen complex phase in the printed notation can be absorbed into the coefficients.
For the specified mass range, $1/2<\nu<\sqrt5/2<3/2$. The physical fluctuation is $\delta\chi_{\mathbf k}=v_{\mathbf k}/a=-H\tau v_{\mathbf k}$, so its two branches behave as $(-\tau)^{3/2-\nu}$ and $(-\tau)^{3/2+\nu}$. Both exponents are strictly positive. Therefore \b[both physical-field branches decay as $\tau\to0^-$]:
$$
\boxed{\delta\chi_{\mathbf k}\propto
\begin{cases}a^{-(3/2-\nu)},&\text{slower branch},\\
a^{-(3/2+\nu)},&\text{faster branch}.
\end{cases}}
$$
The slower branch of $v$ itself grows because $1/2-\nu<0$, but the expanding <scale factor> more than cancels that growth. A spatially homogeneous background field of this mass satisfies the corresponding $k=0$ equation and also has these decaying powers.
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