I don't do CNC (anymore), but I do manually make multi-lead threads. What I am seeing in the picture looks like the offset for the start point is not consistent. With a 7 lead thread, each start is 1/7th the distance of the thread pitch.
For example, with a THREE (3) lead thread which is running at a single pitch of 8 threads per inch, it looks like 24 threads per inch, but has the steepness of 8 threads per inch. Each start Has the Z offset by about 41.66 thousandths (on a manual lathe, the compound is set parallel with the ways, and is the compound handle is cranked to move the cutting tool towards the chuck). This way, you aren't changing your rotational indexing... MUCH easier to do. In your case, the Z offset would be 1/7th of your thread pitch (for a 7 lead thread). What I didn't see clearly stated above, is what your actual actual desired thread pitch is.
Back in the very ancient days of CNC (late 70s and early 80s) we might achieve this by putting in a tool offset in the Z axis for each thread start. Z=Z-(offset value which is 1/7th of the pitch of the major thread).