The infinite nested sumsets conjecture

Let $A\subset\mathbb{N}$ have positive upper Banach density. For an infinite sequence of infinite sets $B_1,B_2,\ldots\subset\mathbb{N}$, define $B_1+\cdots+B_i$ in the usual way.

Infinite nested sumsets conjecture. There is an infinite sequence of infinite sets $B_1,B_2,\ldots\subset\mathbb{N}$ such that

$$B_1+\cdots+B_i\subset A\qquad\text{for every }i\in\mathbb{N}.$$

This strengthens the finite nested sumsets conjecture by requiring one sequence to work for every order. It is posed as an open strengthening.