Thesis The Axial Age is defined by its exclusions, and that makes the thesis unfalsifiable
Where it stands
Every survey of world philosophy I've read includes the Axial Age: between about 800 and 200 BCE, in Greece, India, China and Israel, with no documented contact between them, something similar happened. People began defending claims against stated objections, named individuals disagreed with one another, and inquiry turned to conduct, the self and the grounds of knowledge. Karl Jaspers gave it the name in 1949.
I grew up in Lima. The Andes and Mesoamerica are never in the picture. Nor are Egypt or Mesopotamia, which were not less sophisticated societies. And the reason they're left out isn't dates; it's that they don't fit.
That's my objection. Pick the regions where argumentative philosophy appeared, call the period when it appeared an age, and the pattern is guaranteed. A thesis whose evidence is selected by the thesis can't be tested. Add a six-century window, roughly the distance between Chaucer and us, and almost any set of literate civilisations would yield something.
Jaspers wrote just after the Second World War, looking for a shared human inheritance. That motive doesn't make him wrong. It does mean the thesis was built to have a certain shape.
Thesis: the Axial Age, as an age, is unfalsifiable by construction. Defend it if you can.