Inanna holds Eanna throughout the historical period and there is no doubt about that. The earliest monumental levels are several centuries older than any readable statement of dedication. The tablets recovered from them are accounts and word lists, the buildings carry no dedicatory inscription, and no foundation deposit naming a recipient has been found. The usual identification is an inference backward from later certainty, and no scholar has argued a rival case, because there is nothing to argue from.
What would resolve this A dedicatory inscription, a foundation deposit or an archaic text from the Uruk V or IV levels that names the recipient of the buildings. Failing that, a decisive advance in reading the non-administrative part of the proto-cuneiform corpus, which would let the archaic lists be read as statements rather than inventories.
OpenAssyriology and proto-cuneiform palaeography, working with Near Eastern archaeology of the fourth millennium BCE.Nicole Brisch (ed.), Ancient Mesopotamian Gods and Goddesses