How can biologists rule out multiple early organisms that went through conjugation?
Because they would not use the same DNA codons. Codons are extremely conserved across life, with only minor variations, there is no reason this should be the case unless those codons are inherited from a common ancestor. This is contingent on what you mean by organism, prior to the incorporation of ribosomes it would be more possible.

If what you suggest had occurred it would drastically DECREASE the odds modern organism would share the same DNA.