Regarding my comment about the sentence structure in Greek, you stated, "Not at all.", then added: "(But does the "Spirit of Truth" really care about truth in argument?)"
Truth: Of course truth is important. Of course we need to care about it. And of course the Holy Spirit, the Spirit of Truth cares about truth. By all means, if I have erred, then I stand corrected.
The Watchtower Society (under its Greek man, Fred Franz) was so concerned about this verse, that they considerably altered it by turning "Lord" into Jehovah. Why? Because of the implications in the sentence structure. There is no basis to insert the name Jehovah under any circumstance, but they needed it to read that way. An example in English, (which can be more confusing and duplicitous than Greek,) a sentence might read, "While they were serving him, John spoke up and asked for the salt." In English, "him" would normally be understood to apply to John because his response is in the subordinate clause. However, if one uses sloppy grammar, "him" could be applied to someone else. A good English grammarian would restructure this into two sentences to end the confusion. Of course, I used a pronoun example. Using a proper noun, such as a title, we can achieve the same result: "While they were serving the King, the Highness spoke up and asked for the salt." Here, one would not be able to say that the King is not the same person as the Highness.
As for the classical or Koine' Greek grammar requirements, I obtained that view from reading a commentary on the Holy Spirit which addressed this very issue. I believe I made a note of it in my study paper presented at BRCI ... or at least I noted the resource. I failed to save the resource link, so, I will find my hard copy and make sure I correctly understood it.
Since you clearly assert that the rules of Greek do not require that the "Lord" be tied to the "Holy Spirit" as the sentence would suggest, then maybe you could provide some insight and source information that refutes the point presented.
Here is the Greek at Acts 13:2 as it is commonly found in popular manuscripts: Source Ref. The Polyglot Bible: http://www.sacred-texts.com/bib/poly/act013.htm
I tried to print out the Greek, but JWD font does not support all of the Greek alphabet characters.