In his $G_2$ paper, Kuperberg gives the following numbers of acyclic freeways
for n=0...6: 1, 0, 1, 1, 4, 10, 35. (Which is identical to $dim Inv(V^{⊗n}_{1,0})$, the spanning size of the tangle vector space. (??). But that is NOT
identical with the number of crossingless trivalent tangle graphs:
for n=6, the "hexagon" can't be resolved into the 34 acycling tangle graphs
and must be added to the linear independent set.)
The $G_2$ numbers (or so I think) are identical for the whole $E_7$ family
(except $B_3(\Lambda_3)$, $A_1(3\Lambda_1)$ and $A_1*A_1(\Lambda_1*\Lambda_1)$ where the last number should be 30,34 and 25 if I computed correctly). (But since you now
must use 2 irrep colors for any other than $G_2$, 5 of the 34 above graphs are forbidden.)

With some jump of faith, I suppose also for all members of the $E_8$ family the spanning
space numbers are the same: 1, 0, 1, 1, 5, 15, 70 (???), except for some
"special" groups of that family.

Can you verify my $dim Inv(V^{⊗n}_{1,0})$ values for $E_8$ ? (Maybe with a list of exceptional members ? Directly I can only compute for $A_1(4\Lambda_1)$ and this surely gives less than 70.)