Generalized Iizuka conjecture for successive quadratic fields

Let $n\geq 3$ be an odd integer and let $m\geq 1$ be an integer. For an integer $d$, consider the $m+1$ successive quadratic fields

$$\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1}),\cdots,\mathbb{Q}(\sqrt{d+m}).$$

Generalized Iizuka conjecture. For any odd integer $n\geq 3$ and any integer $m\geq 1$, there is an infinite family of $m+1$ successive imaginary or real quadratic fields of this form whose class numbers are all divisible by $n$.

This strengthens the prime-divisibility formulation by requiring divisibility by every odd integer $n\geq 3$. The supplied text says that the statement is completely proved for $m=1$, while for $m\geq 2$ it discusses only a weaker result; hence the generalized statement remains open in the stated range.