Cobra Kai hasn't directly connected to The Karate Kid 4 yet but it remains a possibility that Julie Pierce (Hilary Swank) could make her presence felt in the hit Netflix series. 1994's The Next Karate Kid was the final on-screen appearance of Noriyuki "Pat" Morita as Mr. Miyagi and it was the first Karate Kid film not to star Ralph Macchio as Daniel LaRusso. Instead, The Next Karate Kid introduced future Academy Award-winner Hilary Swank as Julie, a teenager who became Miyagi's newest apprentice after Daniel-san.
The Next Karate Kid was set in Boston, Massachusetts, all the way across the country from the Los Angeles setting of Daniel LaRusso's Karate Kid trilogy, which also happened a decade prior. In The Next Karate Kid, Miyagi agreed to watch over the troubled, orphaned Julie as a favor to the widow of Lt. Jack Pierce, Miyagi's former World War II Army buddy who was also the earliest known student Miyagi taught karate to. When Mr. Miyagi learned Julie had already learned some of his karate from her father and grandfather, the sensei agreed to continue Julie-san's training so she could curb her grief and anger as well as face the adversaries in her life, which came in the form of a group of ROTC high school bullies called the Alpha Elite.
However, it makes sense that none of those events in 1994 Boston have any bearing on Cobra Kai so far. The hit revival series reignited the teenage rivalry between Johnny Lawrence (William Zabka) and Daniel LaRusso, as well as passing their animosity on to a new generation of teens, including their son and daughter. But even though he's one of the c0-leads of Cobra Kai, Johnny has barely any knowledge of Mr. Miyagi and he has certainly never heard of Julie Pierce. As for Daniel, it's possible his sensei mentioned Julie to him at some point between 1994 and Mr. Miyagi's death in 2011 but it's not clear if they ever met. If they did, it may have happened if Pierce attended Mr. Miyagi's funeral but this interaction hasn't been canonically established by Cobra Kai.
Cobra Kai's producers have stated that they consider every film Mr. Miyagi appeared in to be part of the series' canon, which they informally dubbed the "Miyagi-Verse". So Julie Pierce and the ancillary characters of The Next Karate Kid are potentially in play, but the fact that they are so far removed from the unfolding events of Cobra Kai, which are borne from Daniel's Karate Kid trilogy, makes a crossover tricky and difficult. Still, Julie Pierce is absolutely a part of the greater Karate Kid/Cobra Kai franchise and she is one of the few people who knew and loved Mr. Miyagi and called him her sensei. Of course, Swank is a major movie star, and landing her to appear in Cobra Kai would be a coup but it's certainly not out of the question, especially since Cobra Kai is a massive hit for Netflix, with season 3 and 4 already on the way.
Story-wise, Julie could be a key to the future of Daniel's attempts to honor Mr. Miyagi by reopening the Miyagi-Do dojo and counter Cobra Kai by teaching the Okinawan style of karate to more students than ever before. Cobra Kai season 2 ended in tragedy after an all-out brawl at West Valley High involved Daniel's daughter Samantha (Mary Mouser) and hospitalized Johnny's pupil Miguel (Xolo Mariduena) with life-threatening injuries. Daniel was left at a crossroads as to whether he should continue Miyagi-Do and Cobra Kai season 3 will see LaRusso travel to Okinawa, which teases LaRusso learning secrets Miyagi hid from him.
In this way, Julie Pierce becomes an interesting component of the story because she may know things about Mr, Miyagi that even Daniel doesn't. The Next Karate Kid's events only spanned a handful of weeks so there's plenty of leeway for Cobra Kai to expand upon Julie's relationship with Mr. Miyagi and add numerous, untold developments that could affect, and possibly help, Daniel. Even if Hilary Swank doesn't directly appear as Julie (which would admittedly be disappointing), acknowledging Mr. Miyagi's female protege on the East Coast in Cobra Kai season 3 or 4 could only deepen the overall story and bring balance to the entire Karate Kid movie saga.