The caster uses Blink and appears in the corresponding place in the Ethereal Plane.
Blink (BR 219):
you vanish from your current plane of existence and appear in the
Ethereal Plane (DMG 48):
The Ethereal Plane is a misty, fog-bound dimension. Its "shores,"
called the Border Ethereal, overlap the Material Plane and the Inner
Planes, so that every location on those planes has a corresponding
location on the Ethereal Plane.
If the Immovable Rod (BR 175) is activated it is
magically fixed in place.
this place is corresponding to the same place on the Ethereal Plane in which you, including your possesion, appear.
What happens when the spell ends, and
you return to an unoccupied space of your choice that you can see
within 10 feet of the space you vanished from.
you choose to return to a place that the Immovable Rod is not magically fixed to, is not specified in any of the (D&D 5e) rules that I have read and am aware of.
This question assumes that the Immovable Rod is in the possesion of the character as part of their harness and that is an awkward situation in itself.
The category of Rods has defined proportions (DMG 139):
A scepter or just a heavy cylinder, a magic rod is typically made of metal, wood, or bone. It's about 2 or 3 feet long, 1 inch thick, and 2 to 5 pounds.
That gives you a general idea of how unlikely it is that the Immovable Rod is not noticed or the praticallity of weaving a harness with an integrated Immovable Rod alltogether.
A lot hinges on whether the DM rules that the Immovable Rod is in your possession in the first place, this is true for other kinds of teleportation as well.
This leads to the next point, a magical item does what is in its description.
the rod doesn't move, even if it is defying gravity.
It can be moved by a special check
A creature can use an action to make a DC 30 Strength check, moving
the fixed rod up to 10 feet on a success.
and there is no line preventing teleportaion as a means to displace the Immovable Rod. So if you teleport the Immovable Rod it ends up in a new space where it is:
magically fixed in place. Until you or another creature uses an action
to push the button again, the rod doesn't move, even if it is defying
Furthermore, there is no line that says that the Immovable Rod returns back to places where it was magically fixed whether displaced or moved by a special check. So it doesn't return to a place where it is magically fixed.
On types of movement see PHB 190, pushing, dragging and lifting is covered in the PHB 176.
If the DM rules that the Immovable Rod is not in your possession, then it remains in the place.
An inactive Immovable Rod doesn't prevent movement.