There is a difference of opinion among the “authorities” — Gregg, Morson, CMOS, et cetera — regarding the nature of the word so. Some view it as a coordinate conjunction like and and or. Others view it in the category with therefore and however.
The coordinate conjunctions — and, but, or, nor — do nothing in the language except connect elements that are grammatically equal. That is the definition of a coordinate conjunction. (The word but can be used as a preposition, but this is the only deviation from the basic connecting function.) Since so is an adverb by nature, it would not seem to fit here.
There is another category of conjunction — words that are basically adverbs but can be used as conjunctions. Here is a fairly comprehensive list:
These words start out as basic adverbs, answering the questions when, where, how, why. Occasionally, one is pulled out to the front of a sentence and used as a connecting word for that sentence to the sentence in front of it.
…We missed the train; therefore, we had to drive to the location.
…She started to walk toward the street; then her husband came out of the building directly in front of her.
These words are called “conjunctive adverbs.” They join two sentences and show a relationship between them. They are adverbs by nature that are being used as conjunctions.
So fits here. It is an adverb by nature that is occasionally pulled out to the front of the sentence to make a connection to the sentence in front of it. When it does, it means “therefore.”
Conjunctive adverbs take a period or semicolon in front of them and a comma after if they are more than one syllable.
…He raised his hand slightly; so the doctor believed he could hear him.
…I took the train to San Diego; so I was able to get some work done on the way.