[C++][CI] ASAN error in streaming JSON reader tests

[westonpace]

Describe the bug, including details regarding any error messages, version, and platform.

Example: https://github.com/apache/arrow/actions/runs/3808683807/jobs/6479467470
logs_412266.zip

Component(s)

C++

[pitrou]

cc @benibus

[benibus]

I saw this pop up once back when the threaded reader implementation was a lot more fragile. Figured it was fixed by the subsequent changes but I guess not...

I still haven't managed to reproduce this locally but I'll try to find out what's wrong.