Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \dots, a_n$ all possible values of $a_ia_j - 1$ with $i \ne j$. Is it true that the set of integers which eventually appear has positive density?

Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$. Is it true that the set of integers which eventually appear has positive density?