What do you think will happen to the convention currents in the mantle if the interior of Earth eventually cools down

No convention currents occur.

Explanation:

If the interior part of the Earth is cool down, it will leads to convection currents to stop. Due to this cool down of the earth's interior part cease the motion of the plates and the Earth would become geologically dead. Convection currents within the Earth's mantle occurs when the hot material rising upwards and then dropping back toward the core due to cooling so if the mantle is cool so no movement occurs as well as no convention currents.

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