You'll find both initial caps and initial lower-case:
From Handbook of Landscape Archaeology, B David and J Thomas (eds),
... the linkage between the Arabian Desert in the east and the Sinai
and Saharan Deserts in the west,....
From Great Walls and Linear Barriers by P Spring,
In the semi-arid northern borders of the Arabian and Saharan
I haven't been able to find a conjunction of Saharan with anything but another adjectival form of a place name, but the preponderance of the Ngram viewer examples seems to be lower-case deserts. The probable reason is the the Sahara Desert is a proper name, but Saharan deserts isn't. Rather it's a descriptive term for the various so-called xeric subregions of the Sahara Desert, which have their own proper names, e.g., the Grand Ergs of Algeria.
This discussion illustrates the fact that the adjectival form of a proper noun keeps its properness (and thus its initial cap) for itself, but also that the properness does not extend to phrases that are proper with the noun form. Thus we learn from wikipedia that
The Mississippi Band of Choctaw Indians is one of three Federally
recognized tribes of Choctaw Indians.
But we would say that
The moundbuilding, Mississippian bands of Indians formed an urban and suburban culture that flourished between 800 an 1600.
In the former case Band is part of the name and so is capitalized. In the latter case, band is not part of the name and so is lower-case, unaffected by Mississippian, which simply acts as an adjective no different from its companion, mound-building.