[lkozma]

This is a very nice and deep question. Are you aware of the published research on the problem?

The growth rate is indeed around 4 and now known to be strictly above 4: https://page.mi.fu-berlin.de/rote/Papers/pdf/Lambda-4.pdf

If I remember correctly, there was a non-rigorous argument for a concrete conjectured value not far above 4.

[alcover]

Thanks for the link.

  we obtain 4.00253176 as a certified lower bound

How intringing it's strictly just above 4 !

Maybe it has to do with symmetries. The proportion of symmetric polys in successive generations goes towards zero. I suspect that this 'pollutes' the asymptot.

About published research, I just glanced at the little I found, (as I often can't do more than glance, being limited in Maths.)

Also I prefer to go at it naively first.