  In this paper, we investigate the Bohr radius for $K$-quasiregular
sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk
$\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is
quasi-subordinate to some analytic function. The main aim of this article is to
extend and to establish sharp versions of four recent theorems by Liu and
Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two
conjectures proposed by them. Furthermore, we establish two refined versions of
Bohr's inequalities and determine the Bohr radius for the derivatives of
analytic functions associated with quasi-subordination.
