A query regarding P vs NP

Problem Statement: Given a Problem $ (A)$ such that: Any proposed solution $ (S_0)$ is guaranteed to be polynomial size w.r.t. $ A$ . Now, consider a fixed set of $ K$ properties $ \{k_0, k_1,…k_c\}$ , where each property is $ coNP-Complete$ .

For any solution $ S_0$ is considered a valid solution iff: “None of the property in $ K$ has a $ NO$ instance for $ S_0$ “.

Example: For some hypothetical problem $ (A’)$ let the solution $ S_0’$ be a simple graph with $ n$ nodes. Let $ k_i$ be the property: “$ S_0’$ does not contain a clique with $ n/4$ nodes”.

Query: Assuming the highly improbable scenario that $ P=NP$ , it implies $ P=NP=co-NP)$ . Thus, all the properties in $ K$ are $ P-Complete$ . Does that imply that finding the solution $ (S_0)$ for $ (A)$ is in $ P$ ? It seems so but still need confirmation.