Does the sequence built from $2,3$ by adjoining all $a_ia_j-1$ have positive density? (Erdős #424)

Start with $a_1=2$ and $a_2=3$. Repeatedly enlarge the current set of terms $a_1,\ldots,a_n$ by adjoining all values $a_ia_j-1$ with $i\ne j$; let $A$ be the set of positive integers that eventually appear. The sorted sequence begins $2,3,5,9,14,17,26,\ldots$. Does $A$ have positive lower density — that is, is there a constant $c>0$ such that $\lvert A\cap[1,x]\rvert\ge cx$ for all large $x$? (Note that $A$ cannot have density $1$: if $a,b\in\{0,2\}\pmod 3$ then $ab-1\in\{0,2\}\pmod 3$, so by induction no integer $\equiv 1\pmod 3$ ever appears, giving $A$ upper density $\le 2/3$.)

Acceptance. FULLY RESOLVES: prove that $A$ has positive lower density — exhibit an explicit $c>0$ and prove $\lvert A\cap[1,x]\rvert\ge cx$ for all large $x$ — or prove that $A$ has lower density $0$; a complete proof, machine-checkable or fully written. ADVANCES (each independently checkable): (a) prove any nontrivial lower bound on the counting function $\lvert A\cap[1,x]\rvert$ that is asymptotically larger than any proven bound in the background (which records none beyond triviality), with proof; (b) prove an upper density strictly below $2/3$, or upgrade an upper bound to a proven statement, with proof; (c) compute $A\cap[1,x]$ to a new record $x$ with reproducible code, reporting the empirical density and its trend. Deliver the proof, or the generator code plus the attained $x$ and density estimates.