Alphonse Karr is notorious for attracting aphids. I used to have some that looked just like that (actually worse), and had to continually spray with Safer to keep them under control. I don't believe those are eggs...just aphid byproduct.

I have the SAME issue with my AKs. I have sprayed a few times and they seem to have improved but still not as healthy looking as I'd like. That being said, should I continue to spray until everything is gone?

Yes...keep spraying, preferably with something like Safer that is not a toxic pesticide. Depending on where you are and what your climate is, this could remain a chronic problem which requires regular spraying to keep under control.

I highly recommend the Bayer tree and shrub granules for aphid problems on bamboo. Your plants will thank you and im not super into using chemicals and I tried to avoid that stuff for a few years, once I tried it my plants leaves got bigger more of them, none of the black sut that can develop.... its a winner in my book...