The vertex-deletion conjecture for correspondence packing number

Let $G$ be a graph and let $v\in V(G)$. Vertex-deletion conjecture. Then

$$\chi_c^\star(G\setminus v)\ge \chi_c^\star(G)-2.$$

Equivalently, adding a new vertex adjacent to all existing vertices increases the correspondence packing number by at most two. This would formalise the claim that no vertex contributes more than two to correspondence packing; the supplied text gives no resolution status.