As far as I can see, the only link between Marsh Arabs and Sumerians presented in this paper comes from dating Y-STR variation of their major J1-Page08 group using the evolutionary mutation rate, with a divergence time of 4.5 +/- 2.6 ky. Even if that mutation rate was correct (it is not) and the assumptions on which the confidence interval are based were exhaustive (they are not), we still have +/- 2.6 ky leeway to deal with, which spans not only the Sumerians but plenty more besides.
Not to mention that the evolutionary mutation rate is wrongly applied to every case under the sun, and that Y-STR based age estimation in general has been conclusively shown to be a rather futile exercise.
Nonetheless, the paper does have value in demonstrating the paucity of J2 and R1 in the Marsh Arabs compared to the more cosmopolitan general Iraqi population:
Rather than "Sumerian", it seems that the Marsh Arabs have rather preserved a more pristine Semitic patrilineal gene pool compared to the cosmopolitan Iraqi samples that have absorbed pre-Arab and pre-Semitic population elements.
Different from the Iraqi control sample, the Marsh Arab gene pool displays a very scarce input from the northern Middle East (Hgs J2-M172 and derivatives, G-M201 and E-M123), virtually lacks western Eurasian (Hgs R1-M17, R1-M412 and R1-L23) and sub-Saharan African (Hg E-M2) contributions.
BMC Evolutionary Biology 2011, 11:288doi:10.1186/1471-2148-11-288
In search of the genetic footprints of Sumerians: a survey of Y-chromosome and mtDNA variation in the Marsh Arabs of Iraq.
Nadia Al-Zahery et al.
For millennia, the southern part of the Mesopotamia has been a wetland region generated by the Tigris and Euphrates rivers before flowing into the Gulf. This area has been occupied by human communities since ancient times and the present-day inhabitants, the Marsh Arabs, are considered the population with the strongest link to ancient Sumerians. Popular tradition, however, considers the Marsh Arabs as a foreign group, of unknown origin, which arrived in the marshlands when the rearing of water buffalo was introduced to the region.
To shed some light on the paternal and maternal origin of this population, Y chromosome and mitochondrial DNA (mtDNA) variation was surveyed in 143 Marsh Arabs and in a large sample of Iraqi controls. Analyses of the haplogroups and sub-haplogroups observed in the Marsh Arabs revealed a prevalent autochthonous Middle Eastern component for both male and female gene pools, with weak South-West Asian and African contributions, more evident in mtDNA. A higher male than female homogeneity is characteristic of the Marsh Arab gene pool, likely due to a strong male genetic drift determined by socio-cultural factors (patrilocality, polygamy, unequal male and female migration rates).
Evidence of genetic stratification ascribable to the Sumerian development was provided by the Y-chromosome data where the J1-Page08 branch reveals a local expansion, almost contemporary with the Sumerian City State period that characterized Southern Mesopotamia. On the other hand, a more ancient background shared with to Northern Mesopotamia is revealed by the less represented Y-chromosome lineage J1-M267*. Overall our results indicate that the introduction of water buffalo breeding and rice farming, most likely from the Indian sub-continent, only marginally affected the gene pool of autochthonous people of the region. Furthermore, a prevalent Middle Eastern ancestry of the modern population of the marshes of southern Iraq implies that if the Marsh Arabs are descendants of the ancient Sumerians, also the Sumerians were most likely autochthonous and not of Indian or South Asian ancestry.