From the provided statements regarding the Sahara Desert, it seems that all of them are generally accurate with one possible exception. Let's evaluate them:
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The Sahara is the largest desert in Africa. - True. The Sahara Desert is indeed the largest hot desert in Africa.
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It is the homeland to the largest desert on the globe. - This statement is somewhat misleading. While the Sahara is the largest hot desert, Antarctica is the largest desert overall (cold desert). So, this statement could be considered false.
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The Sahara extends into southern Africa. - Partially True. The Sahara primarily stretches across northern Africa rather than southern Africa, although it does extend into some parts of the Sahel region, which can be considered southern relative to the Sahara itself.
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The region stretches across one-fourth of the African continent. - True. The Sahara does cover a significant portion of Africa, roughly about one-fourth of the continent.
Based on this analysis, the statement "It is the homeland to the largest desert on the globe" is likely the one that is except or false.