OK, let's run my argument again from a different starting point - by considering a septuple Fibonacci, bearing in mind that examples of the double Fibonacci have already been observed. This would normally go: 0 7 7 14 21 35 56 91 147 . . . This means we don't have to allow ourselves a "close" since 56 is already perfectly correct here. I then allow myself just one other tweak, that is swapping round the 21 and 35, so that as long as we stick with the same recurrence relation, at least in a rightwards direction, the sequence now goes . . . 35 21 56 77 . . . The next term would be 133, though admittedly it doesn't look like sunflower heads ever run to this quantity of spiral arms.
The other two Fibonacci generalisations considered in the article, Lucas and F4, can also be regarded as the result of swapping round two terms in the original Fibonacci sequence and adding them together. In Lucas those are the 2 and the 1, in F4 they're the 3 and the 1, going 3 1 4 5 9 etc. Why not swap round terms further along?
I'm also very interested in those Fibonacci generalisations with different recurrences, such as Narayana's Cows (OEIS A000930), and wonder if they too crop up in natural structures and processes such as sunflower seed heads. Projects such as yours are obviously very important for this, and I will study your paper closely.