Does Tiny Hut protect from Dream?

Leomund’s Tiny Hut spell states

Spells and other magical effects can’t extend through the dome or be cast through it.

Dream states

Choose a creature known to you as the target of this spell. The target must be on the same plane of existence as you.

Can Dream reach a target sleeping in Leomund’s Tiny Hut?

The points giving me doubt are:

  • "Extend through the dome" seems to exclude the floor/ground.
  • Does "Target must be on the same plane of existence" override any protection offered by the dome, or is it just a requirement of Dream, and doesn’t override any other existing limitations?