Let $\epsilon > 0$. Is there some set $A\subset\mathbb{N}$ of density $> 1 - \epsilon$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?

Let $\epsilon > 0$ be given. Then, for a sufficiently large prime p, take the sequence of consecutive primes $p_1 < \cdots < p_k$ such that $$\sum_{i=1}^k \frac{1}{p_i} < 1 < \sum_{i=1}^{k + 1} \frac{1}{p_i},$$ and let $A$ be the set of all naturals divisible by exactly one of $p_1, ..., p_k$ (with multiplicity $1$). Then $A$ has density $\frac{1}{e} - \epsilon$ and has the property that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$.