Limit-point conjecture for normalized sum-product pairs

Let $\operatorname{NSPP}([3,\infty))$ denote the multiset of normalized sum-product pairs for positive-integer sets of cardinalities at least $3$, with each pair lying in $[1,2]^2$. The relevant sum-product coordinates are normalized so that the extremal value $2$ corresponds to quadratic-scale growth. Limit-point conjecture for normalized sum-product pairs. The limit points of

$$\operatorname{NSPP}([3,\infty))$$

are confined to the two line segments

$$\{2\}\times[1,2]\quad\text{and}\quad [1,2]\times\{2\}.$$

This is the normalized geometric formulation of the Erdős sum-product conjecture: asymptotically, normalized pairs should move north and east. The source gives no proof or disproof of this formulation.