  The center conjecture for the cyclotomic KLR algebras $R_\beta^\Lambda$
asserts that the center of $R_\beta^\Lambda$ consists of symmetric elements in
its KLR $x$ and $e(\nu)$ generators. In this paper we show that this conjecture
is equivalent to the injectivity of some natural map
$\bar{\iota}_\beta^{\Lambda,i}$ from the cocenter of $R_\beta^\Lambda$ to the
cocenter of $R_\beta^{\Lambda+\Lambda_i}$ for all $i\in I$ and $\Lambda\in
P^+$. We prove that the map $\bar{\iota}_\beta^{\Lambda,i}$ is given by
multiplication with a center element $z(i,\beta)\in
R_\beta^{\Lambda+\Lambda_i}$ and we explicitly calculate the element
$z(i,\beta)$ in terms of the KLR $x$ and $e(\nu)$ generators. We present an
explicit monomial basis for certain bi-weight spaces of the defining ideal of
$R_\beta^\Lambda$ and of $R_\beta^\Lambda$. For
$\beta=\sum_{j=1}^n\alpha_{i_j}$ with $\alpha_{i_1},\cdots, \alpha_{i_n}$
pairwise distinct, we construct an explicit monomial basis of
$R_\beta^\Lambda$, prove the map $\bar{\iota}_\beta^{\Lambda,i}$ is injective
and thus verify the center conjecture for these $R_\beta^\Lambda$.
