GX 339-4 is in outburst. We will be observing it with JWST on September 10 from 00:42:23 to 3:03:17 UT. It's currently about 16.5 mag in V, and it's likely to be getting brighter.
The goal of the JWST program is to look at the rapid variability, so sub-second timescale time series photometry would be especially valuable for anyone who has that capability, but any time series photometry with accurate timestamping would be useful.
Also, there is a bit more detail under the time sensitive alerts about what we are trying to do, but please go as fast as you can without being dominated by read noise, even if that means that the individual exposures might have low signal to noise.