  Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and
$\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely
many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$,
$\mathbb{Q}(\sqrt{d+1})$ and $\mathbb{Q}(\sqrt{d+k^2})$ with $d \in \mathbb{Z}$
such that the class number of each of them is divisible by $3$. This
affirmatively answers a weaker version of a conjecture of Iizuka
\cite{iizuka-jnt}.
