Idempotence conjecture for the lift of a partial field

Let $\mathbb{P}$ be a partial field, and let $\mathbb{L}\mathbb{P}$ denote its lift. Lift idempotence conjecture.

$$\mathbb{L}^2\mathbb{P} \cong \mathbb{L}\mathbb{P}.$$

The conjecture is related to whether the canonical homomorphism from the lift back to the original partial field induces a bijection on fundamental elements. The paper gives no resolution.