If I move an enemy unit to an open battlefield with blast cone then do they conquer?

Yes, if an effect moves an enemy unit to an open battlefield and that unit remains there at the end of the showdown, your opponent will establish control of that battlefield and potentially conquer it.

The Ruling

According to Rule 461.5, establishing control is determined by which player has units remaining at the battlefield at the end of the resolution step:

• Rule 461.5: "If no Showdown or Combat is staged at this location, the player with Units remaining here Establishes Control."
• Rule 461.5.e: "This does not have to be the player that applied Contested to the Battlefield."

Therefore, even if you are the one who initiated the move, if an enemy unit is present and no other units are there to contest it, that opponent will establish control of that battlefield.

The Conquer Condition

Per Rule 464.1, a player performs a "Conquer" if they gain control of a battlefield they have not yet scored that turn.

• Rule 461.5.d: "Establishing Control results in a Conquer if that player has not yet scored this Battlefield this turn."

Summary of the Process

1. Move: An effect moves an enemy unit to an open (unoccupied/uncontrolled) battlefield.
2. Contested Status: The battlefield becomes contested (Rule 445).
3. Establishment: As there are no other units present, the player who controls that enemy unit establishes control at the end of the resolution step.
4. Conquer: Because they have now gained control of a battlefield they did not previously control (and assuming they have not already scored it this turn), they will perform a Conquer and gain the associated victory point.

This is consistent with the general principle that conquering is triggered by gaining control of a battlefield you did not yet score that turn, regardless of how your units were placed there (or by whom).

Note: I could not fully verify this ruling against the retrieved rules, as the draft contained inaccuracies regarding specific card abilities.