One of the hypotheses that can be tested better than ever by JWST is precisely this: Is the very-distant universe homogenous?

Modern cosmology assumes-so, but it need not be true.

Strongly agree, except that "assumes" can be unpacked a bit. There are various reasons why cosmologists think the distant universe is homogeneous, and some of them (not all) are closely based on evidence, most notably the approximate homogeneity of the cosmic microwave background, which is (in a sense) further away than anything the JWST can see.

JWST is not particularly well suited to testing homogeneity. Survey telescopes that observe a large fraction of the sky are much better suited to that task.

Homogeneity has been very well tested out to a redshift of a few by surveys like the Sloan Digital Sky Survey and the Dark Energy Survey, each of which have observed thousands of square degrees of sky (there are about 41k square degrees in total). And as another commenter notes, the CMB (at redshift 1100) is homogeneous (to one part in 10k).

We would like that to be true, but it seems like it might not be.

https://en.wikipedia.org/wiki/Axis_of_evil_(cosmology)

https://en.wikipedia.org/wiki/CMB_cold_spot