The multiplicative bounded-product conjecture

Let $A\subset\mathbb{N}$ have positive density with respect to a multiplicative Følner sequence. For an infinite set $B\subset\mathbb{N}$ and $t\in\mathbb{N}$, consider

$$\{b_1b_2t:b_1,b_2\in B,\ b_1\ne b_2\}.$$

Multiplicative bounded-product conjecture. There exist an infinite set $B\subset\mathbb{N}$ and $t\in\mathbb{N}$ such that

$$\{b_1b_2t:b_1,b_2\in B,\ b_1\ne b_2\}\subset A.$$

This is presented as the multiplicative analogue of the positive-density two-fold sumset problem. The source explicitly says it is open.