My suspicion (and I will admit that at this point it is only a suspicion) is that Rita may have been illegitimate (that is, that Ralph Geissert may have never actually married to Salome), and that as a result any mention of Rita was studiously avoided in later sources from these Geisserts. Note that the only source giving the last name "Geissert" for Salome is Rita's 2000 obituary - which is strange, because at that point Salome should have been referred to either as Salome Bursch (her last married name, which was used in Salome's own obituary) or as Salome Buchunas (her maiden name).
The LDS has filmed (but not indexed) the marriage licenses for St. Clair County, IL (in which East St. Louis is located) through 1941, but these records can only be accessed at a Family History Center or Affiliate Library (https://www.familysearch.org/search/catalog/960691). The St. Clair County Genealogical Society, however, has published an index to these marriage records in their Quarterly, with the indexes for 1931-1934 in their vol. 39, published in 2016, and indexes for 1935-1938 in their vol. 40, published in 2017 (see https://stclair-ilgs.org/quarterly-vols-36-40/). The very comprehensive surname index for the Quarterly, however, does not include a hit for Geissert or Geisert in vols. 39 or 40, indicating that Ralph Geissert did not get married in St. Clair County (where both he and Salome were living) in the early 1930s.
The ultimate proof, of course, would come from Rita's birth certificate - but it appears the St. Clair County Clerk does not issue records for genealogical inquiries.
Lacking that, autosomal DNA is probably the best avenue for establishing any kind of proof here.