I'm going to take a stab at answering this, although I hope someone comes along and writes an even better answer. To begin, I agree with Tᴚoɯɐuo that there is an error in the source material. It may not be entirely ungrammatical, but it's completely nonidiomatic to use "do so" to refer to "having." I'll return to this later.
Your question is whether you can leave out "not" and the answer to that is "no." Did so has a positive polarity. This is incorrect:
He did not answer the question at the time, and did so until ten years later.
You must repeat the negation to match the polarity:
He did not answer the question at the time, and did not do so until ten years later.
Your example in the comments is also incorrect:
I had no pocket money when I was a child and did so until I was well past 17.
This has both the polarity problem and the "can't use do in place of have" problem. If you use "have" as the main verb, a native speaker just wouldn't usually say "do so," except in error. This is downright unnatural sounding:
I did not have a banana yesterday and did not do so until I bought one today.
With "have" we repeat the verb instead of using "do" as a stand-in (repeating the negation is still required):
I did not have a banana yesterday and did not have one until I bought one today.