On an indivisibility version of Iizuka's conjecture

Iizuka’s conjecture predicts that, given $m\in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of $\mathit{\text{all}}$ of the following quadratic number fields, $$\mathbb{Q}(\sqrt{n}),\mathbb{Q}(\sqrt{n+1}),\dots ,\mathbb{Q}(\sqrt{n+m}),$$ are divisible by $p$. In this article, given $k$ and $m$, we study the proportion of $n$ such that the class numbers of $\mathit{\text{none}}$ of the successive fields $$\mathbb{Q}(\sqrt{n}),\mathbb{Q}(\sqrt{n+1}),\dots ,\mathbb{Q}(\sqrt{n+m}),$$ are divisible by $3^k$. Moreover, we study the proportion of imaginary biquadratic fields whose class numbers are not divisible by $3$.