Crossing-change invariance conjecture for lifts of tree-type webs

Let $\tilde{G}$ and $\tilde{G}'$ be tangled tree diagrams related by crossing changes, and let $G\in\mathsf{Tree}_{\mathfrak{sp}_4,\Sigma}$ be a tree-type elementary web. Crossing-change invariance conjecture. If $\tilde{G}$ is a lift of $G$, then $\tilde{G}'$ is also a lift of $G$. The claim would make the lift property independent of the chosen crossing data within a crossing-change class; the source gives no resolution.