  Lin introduced the partition function $\text{PDO}_t(n)$, which counts the
total number of tagged parts over all the partitions of $n$ with designated
summands in which all parts are odd. For $k\geq0$, Lin conjectured congruences
for $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo
$3^{k+2}$. In this article, we develop a new approach to study these
congruences. We study the generating functions of $\text{PDO}_t(8\cdot3^kn)$
and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+3}$ for certain values of $k$. We
also study $\text{PDO}_t(n)$ modulo powers of $2$. We establish infinitely many
congruences for $\text{PDO}_t(n)$ modulo $8$ and $32$. We prove several
congruences modulo small powers of $2$ and discuss the existence of congruences
modulo arbitrary powers of $2$ similar to those in Lin's conjecture. In
reference to this, we also pose some problems for future work.
