Given that John and Julie play on the same level and have 50/50 chance of winning any particular game, I agree with Bonanova that Julie has 50% chance of winning whatever series they choose to play. However, I think the OP is asking about the probability of Julie winning with the score 4:3 in the 7 game series vs. the probability of winning 5:4 in the 9 game series. Those probabilities are different...
Looks like you're right, BMAD. I haven't taken my 9-long worm example far enough to see that the adversary cannot continue to maintain that pattern indefinitely as long as surgeon does the right thing.
1) draw 3 additional lines parallel to each cevian passing through the remaining vertices of 2 triangles.
2) show that 6 other shaded triangles are mirror images of the green triangle and therefore have the same area
3) show that every part of a shaded triangle that lies outside of the big triangle has an equal unshaded part inside the big triangle. Midpoints of big triangle's sides are points of symmetry.