= Solution
<Conformal time> is defined by
$$
d\tau=\frac{dt}{a(t)}.
$$
For $a=a_0e^{H_{\rm inf}t}$, choose $\tau\to0$ as $t\to+\infty$. Integration gives
$$
\boxed{
\tau=-\frac{e^{-H_{\rm inf}t}}{a_0H_{\rm inf}}
=-\frac1{aH_{\rm inf}},
\qquad -\infty<\tau<0
},
$$
as recorded by <Conformal time during de Sitter expansion>.
Writing a prime for $d/d\tau$, the relations
$$
\frac d{dt}=\frac1a\frac d{d\tau},
\qquad
\frac{a'}a=-\frac1\tau
$$
turn the Fourier-mode equation into
$$
\widehat\phi_{\mathbf k}''
+2\frac{a'}a\widehat\phi_{\mathbf k}'
+c^2k^2\widehat\phi_{\mathbf k}=0.
$$
Set
$$
\widetilde\phi_{\mathbf k}
=a\widehat\phi_{\mathbf k}
=-\frac1{H_{\rm inf}\tau}\widehat\phi_{\mathbf k}.
$$
A direct substitution gives
$$
\widetilde\phi_{\mathbf k}''
+\left(c^2k^2-\frac{a''}a\right)
\widetilde\phi_{\mathbf k}=0.
$$
Here $a''/a=2/\tau^2$, so the <Canonically rescaled de Sitter scalar mode> obeys
$$
\boxed{
\frac{d^2\widetilde\phi_{\mathbf k}}{d\tau^2}
+\left(c^2k^2-\frac2{\tau^2}\right)
\widetilde\phi_{\mathbf k}=0
}.
$$
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