This is an interesting question with multiple answers.
HBV cure research never really stopped, but it slowed to just a trickle for a long time. The slowdown started in the late 80’s or early 90s after the wonderful vaccine was approved. There became an assumption at the US NIH (and presumably other major funders too) and the biotech/pharmaceutical companies that the disease would just fade away with time, so research funding got very difficult to obtain in many places. About the same time HCV was discovered and a lot of money was poured into understanding it, so many HBV researchers shifted all or part of their focus to HCV (I worked on both HBV and HCV for most of the time between 2000 and 2012). Thankfully, that effort vs. HCV was stunningly successful. So the core of the problem was that shade was thrown on HBV research by the vaccine and the urgency to address HCV. The final issue is that HBV is going to be harder to cure than HCV was due to its replication mechanism (the nuclear “cccDNA” form of the HBV genome needs to be eliminated, but that is a really hard target). The experimental systems for HBV were (and still are) quite limited in some aspects of what we need to study to develop a cure, and that makes everything even harder.
The very good news is that there is a massive, world-wide academic, biotech, and major pharma effort ongoing to develop a cure for HBV. The number of scientists pursuing cure projects is very large, and good progress is being made. Part of the credit for the very rapid scale-up vs. HBV ironically comes from HCV research. Major pharma had developed truly impressive research capacity for hepatic virology, and so they just refocused a large number of talented people and funding towards HBV. The shift in industrial resources towards HBV was really quite breathtaking in its scale and how fast it occurred!
I’ll end with this note of optimism: We cannot tell when cure(s) will become available because it is impossible to predict the pace of scientific research. However, there are so many cool projects being pursued that some of them will certainly succeed. I feeling is that cure rates will start rising from their current ~5-7% rate within a few years, and that probably within a decade or so we will be routinely curing at least half of the patients entering treatment–maybe more!!! Rest assured, the HBV research community won’t rest till cure rates for HBV match those for HCV.