About an equation in page 8

[starsong98]

Hello, thank you for your amazing survey paper.

In page 8, there is this expression:

"This yields a sequence of noisy data densities $q(x_1), q(x_2), ... , q(x_T)$, where $q(x_t):=\int q(x_t)q(x_0)dx_0$."

Should the latter expression not be about conditional densities? That is, shouldn't it be:

$q(x_t):=\int q(x_t|x_0)q(x_0)dx_0$

Thank you.