  Let $k, t$ be coprime integers, and let $1 \leq r \leq t$. We let
$D_k^\times(r,t;n)$ denote the total number of parts among all $k$-indivisible
partitions (i.e., those partitions where no part is divisible by $k$) of $n$
which are congruent to $r$ modulo $t$. In previous work of the authors, an
asymptotic estimate for $D_k^\times(r,t;n)$ was shown to exhibit unpredictable
biases between congruence classes. In the present paper, we confirm our earlier
conjecture that there are no "ties" (i.e., equalities) in this asymptotic for
different congruence classes. To obtain this result, we reframe this question
in terms of $L$-functions, and we then employ a nonvanishing result due to
Baker, Birch, and Wirsing to conclude that there is always a bias towards one
congruence class or another modulo $t$ among all parts in $k$-indivisible
partitions of $n$ as $n$ becomes large.
