Bolded for emphasis. That's what I've always said. The only final closure is death. As long as characters are alive, it's always possible to ask "what happened then"? For the same reason, I often propose that people stop asking, lest the writers listen too closely and make the characters die.
Having said that, with Trespasser the Inquisitor/Solas romance screams for some kind of continuance. After the original ending, it had ended in a defined state of death (of the romance, not the characters of course), but Trespasser re-opened the story between them. It should be, however, a story we watch, or a story we may be able to influence from the outside with the next protagonist, rather than being one of the participants. I see the choice at the end of Trespasser as a guideline from the player to the writers, as for how they should write her as an NPC in scenes related to Solas.
Personally, I think that prior to ME3, Bioware would have had Trespasser end with the choice of the Inquisitor dying or relinquishing the power of the Anchor to Solas (making him more powerful, so potentially the morally "wrong" choice). Since ME3's ending, Bioware has had to play it very safe with their characters and storytelling (see non-threatening Corypheus, no mentionable deaths for much of DAI, an "easy" out in the Fade by sacrificing Stroud, etc etc). So they didn't want to even have the possibility of killing the Inquisitor for fear of fan backlash. I still see people posting about how morally wrong it was to perform the dark ritual, and if they didn't, they or Alistair would have to die. Yeah I get it, it's a crappy situation, that's the whole point. But now, Bioware is in a position where they can't kill off the Inquisitor but have to have some reasonable way to prevent them from returning as PC. Hm, depower them, have the Inquisition disbanded or under the control of the Divine, mention that they need new agents outside of their chain of command in order to hunt Solas. I guess that's not clear enough that they aren't coming back as a PC.
Granted, I understand that if Inky appears as an NPC in the next game, there's a big risk of that appearance failing like Hawke did in DAI. And yes, the Inquisitor has a much stronger relationship to Solas than a new protagonist will. But unfortunately, that is how the situation has been written and it's a matter of making the best out of a suboptimal situation.
Antagonistic Inquisitors swore to kill Solas. They were manipulated by him for a year. He destroyed their arm, and then told them to their face that he was going to kill them. Solas himself refuses to tell the Inquisitor what his plans are because that would be giving the Inquisitor a weapon to use against him. Everything in that scene says the Inquisitor is going to stop Solas. None of that requires a positive relationship.
And none of that requires the Inquisitor to be the PC. There are other ways to fight a shadow war than to be on the front lines. I would argue that being on the front lines is probably less effective than being an opposing shadow leader directing the cold war.
A small cameo of the inquisitor even just to have closure specially for Solasmancers is ok. Also having the inquisitor back as protagonist brings complications. Fans would want their LI with them on the new game. And since there are at least 6 or 7 romancable characters in DAI.... ..then we can just forget getting fresh new companions. It will be DAI ;02.
I wouldn't mind if the PC in DA4 fights through Solas' forces, finally reaches the boss level, and then has to sit back and watch as the Inquisitor walks in and deals with Solas at the end. Not the best storytelling or gameplay in my opinion, but it's a realistic method of bridging the narrative gap from DAI.
And good point with the LIs. That is a big draw for a majority of people posting on this forum. If we either have them off screen like Hawke's LIs in DAI, kill them off and get new options, or just don't mention them, people will be upset.
I didn't want the Warden back, I think DAI is the best Bioware game since DAO, and I don't want the Inquisitor back as the protagonist. I don't think I'm in a minority - should I make more threads to balance the general impression? 
BTW, just because a thread named "Why is DAI a failure" (0 likes for the OP) remains on the first page, that doesn't mean the general impression is bad. First reply: "Excuse me. I thought DAI is mind-blowing" (100 likes).
Also, it's perfectly possible to like a character and their story and still to prefer not to see them again as the protagonist of a new game. For instance, if really liked the ending, then future instalments with the same protagonist are more likely to disappoint. My Warden's ending in DAO was perfect, so I'm glad they didn't given him a major role in DAI. In the Inquisitor's case, the new ending did disappoint as opposed to the original one, but any new story would be very unlikely to balance that out, given the source of my disappointment, so they'd better be gone as long as my annoyance level remains in non-ME3 bounds. A fresh start in Tevinter is just what I need.
+1