Proto-writing before proto-Sumerian
In the paper Humans 40,000 y ago developed a system of conventional signs by Christian Bentz and Ewa Dutkiewicz I think the lunar calendar hypothesis is very likely for at least some of the artefacts. In figure 1A you can see that the number of vertical indentation marks is either 12 or an adjacent number, like 11 or 13, and then you have 5 marks at the top and bottom. This could be an object carried around to keep track of the month of the year (every time there is a full moon you make another indentation), with some significant event happening either on the fifth month or perhaps the fifth day of each month.
Another interpretation for the 1A object is that the adorant on one side with legs and arms splayed out might represent a baby on its back, and that the indentations are a way of counting down to when a baby is born, or it could be a mother's way of keeping track of the exact age of a new child in years and months (in this case up to 6 years old).