While Sol might be a political and production powerhouse, the GTVA has about 20 or so star systems and combined power of 2 races. Sol fighting against the GTVA? That would be not only terribly stupid but also suicidal.
Realisticly, Sol has no real chance in a war, it's simply outclassed in manpower, industry and versatility. It might be able to put up a good fight, but that's about it.
I wouldn't bank on that. The relationship between Sol and the rest of the systems of the GTA was never made clear. It could be that Sol was simply the most powerful of a relatively equal alliance of star systems. Or it could be that Sol was the powerhouse of the GTA with the others merely being colonies.
I seem to remember reading that Subspace was only discovered somewhere between 30-50 years before the start of the Great War but I can't remember where now. If true that probably wouldn't be enough time for the colonies to reach anywhere near the power of Sol.
Ten years after the end of The Great War the Luyten New Alliance was a major power. Luyten was uninhabited at the time of the great war. If a system can go from nothing to regional superpower in 10 years it doesn't say much for the power of the other regions. Yes some of that is going to be due to the rest of the former GTA falling apart after the disconnection from Earth but it's a pretty big stretch of the imagination to believe that powerhouse systems of the GTA could fall that far that fast.
Besides it only shows that if Sol didn't fall apart it would have a massive head start over the other former GTA regardless.
So yes, not only do I believe it's plausible that Sol could have the might to take on the GTVA and fight a protracted war similar to that the NTF did. You're the one who always likes to claim that a proper defence can hold any node, are you now going to claim that Sol couldn't hold their node against the GTVA using the techniques you're constantly claiming are so effective?
But I doubt that a defensive war is all Sol is capable of, I'll go further and say it's not unreasonable to say that if it came to a war they might be able to wipe out the GTVA fleet - especially if the Vasudans didn't get into it.