The product of two lognormal distributions may not be a lognormal If two lognormal random variables $X, \:Y$ are neither independent nor jointly normally distributed, the product $XY$ may not be lognormally distributed.
Where can I find the explanation for the 'may not be' part in the second statement?

Suppose $W$ and $X$ are independent and lognormally distributed, each with median $m$. Let
$$Y = \begin{cases} \min(W, m/W) \text{ if } X\le m\\
\max(W, m/W) \text{ if } X>m \end{cases}$$
Then $X$ and $Y$ are both lognormal, and either $X$ and $Y$ are both above $m$ or both below $m$. So $XY$ is bimodal and not lognormal.
