As the opening scene of my Dogs in the Vineyard game, I ran the "an Axe murderer tries to kill you in your sleep" conflict. The conflict went fairly well, though the other two at the table were less engaged. What's the best way to engage non-participating players in a conflict that doesn't involve them?
I submit that in Dogs in the Vineyard, every conflict reasonably engages every player, even if every character is not involved. Here's why:
Every conflict reveals information about the situation the Town is in, and its people.
Every conflict reveals information about the moral fiber of the involved characters.
Every conflict reveals information about the moral fiber of the involved players.
If players aren't paying attention to each other's scenes, something else is wrong. Either they are not connecting to their own characters or the situation in Town doesn't challenge them (escalate, escalate, escalate) or they're just not into the RPG.
I suspect, in your situation, jumping into the Axe Murderer scenario as the first scene didn't hook the other players on the town. Why would it? They aren't involved at all yet. I'm pretty sure the rules encourage or even tell the GM to start every Town more or less the same way: Dogs riding in and Townspeople coming out to greet them and tell them What's Wrong and What They Want. The game hasn't really started till that happens.
An Axe Murderer opening is a grabby structure for a movie, but it's less grabby for uninvolved players. I'll go out on a limb here and guess that you were trying to be a clever storyteller. Stop. You've done your prep by creating the Town. Play the Town and let the players tell the story.
One way is: run conflicts for things the whole group cares about. If the conflict is "Does the axe murderer kill you in your sleep?" (or similar), then only one Dog really cares about the outcome. If it's "Does the Steward leave town?", then, even if one Dog takes the lead, everyone should care about the outcome.
Other than that, however, it can be difficult to engage other people in Dogs conflicts. They work best, I find, when they're just one-on-one.
So keep them short. Sometimes, give in the conflict, in order to keep things short. Give everyone a turn: if I've just had an extended conflict, let someone else have a turn next time. And don't run conflicts for everything: if, for example, only one Dog cares about the outcome, then say yes rather than rolling the dice.
I would suggest you do not. I have run games like Nobilis with players that are mostly non-participating. Some of them are here for the journey and like to watch what others do stuff. Some are really only there to play with their friends and it is more of a social event for them. The best middle ground is to give them a moment where they can engage, that is where there character could take center stage but do not press them to take it.