The article you reference is behind a paywall, but I believe I've found the study they're referencing anyhow. You can find it here for free.
To directly answer your question: no one can really answer that right now. That's not to undermine your question. Any population can go extinct when put in certain conditions.
The authors of the paper call for more controlled research,
Experiments in captivity that replicate a range of demographic
scenarios could help improve our understanding of the process of
cultural song erosion and its impacts on fitness and population growth
[42].
I know that's rather unsatisfying, but it's straight from the horse's mouth. That's science for you. But I'll try to entertain you for a bit, because what would a good answer be without a bit of supposition? As I'm sure you've read, it isn't that the birds aren't singing, it's that they're not singing the right song. The authors note
Regent honeyeaters exhibit strong conspecific attraction, once roaming in flocks of hundreds [23]. However, at least four of the interspecific singers were more than 100 km from the nearest known male at the time they were detected.
Which means when they sing the right song, they do a darn good job of recognizing each other. But unfortunately, these guys are too far apart, and as their population density decreases, they're more likely to learn songs from interspecifics (other species). The authors also measured the ability for birds in captivity to learn their own song and they suspect it could cause problems upon release. A scientist teaching a bird to sing might not be a solution either. Sure, we could just stick a speaker out in the woods and let it go all day, but it's kind of like putting a bandage on a sinking ship.
The real issue here is conservation of these guys' habitats, which involves a variety of processes beyond a boombox in the woods. There are tons of questions we still need answers to. For example, if a population decrease results in young birds learning the wrong songs, what causes the population to decrease in the first place? It's a sort of chicken or the egg question. Or, what if we just take the birds and combine their populations? Well, that might be more harm than good. What causes these birds to learn the songs of other species? Are they too smart for their own good? How heritable are the genes that encode these important songs? So on and so forth, you understand.
For more about conservation, and what can be done, check out an example of the Florida panther and what's been done to protect them. Of course, every group is unique in their needs, but just to get an idea of what it takes to conserve a species.