= Solution
Let
$$
d_t=\#\{i:X_i=t\},
\qquad
r_t=\#\{i:X_i\geq t\}
=n-\sum_{s<t}d_s.
$$
Thus $d_t$ patients recover during month $t$, while $r_t-d_t$ survive that month without recovery. The <likelihood function> is
$$
\boxed{
L(q_1,\ldots,q_T)
=\prod_{t=1}^T
q_t^{d_t}(1-q_t)^{r_t-d_t}}.
$$
The independent $\operatorname{Beta}(T,1)$ prior densities are proportional to $q_t^{T-1}$. <Beta-binomial conjugacy> therefore leaves the coordinates posteriorly independent, with
$$
\boxed{
q_t\mid X
\sim\operatorname{Beta}
\bigl(T+d_t,\ 1+r_t-d_t\bigr)}.
$$
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