Tate shifted symplecticity of multiplicative connection moduli

Suppose $C$ is either an elliptic curve or $\mathbb{CP}^1$ with a fixed framing at $\infty$. Let $G$ be the group used to define the moduli stacks, and let $\Lambda_i$ be the stabilizer of a closed point of $\operatorname{Bun}_G(\mathbb{B})$. The Tate shifted-symplecticity conjecture.

1. The stack $BLG$ is Tate 2-shifted symplectic, and both $BL^+G \to BLG$ and $\operatorname{Bun}_G(C \ \backslash\ D) \to BLG^k$ are Tate 2-shifted Lagrangian.
2. The stack $\operatorname{Bun}_G(C \ \backslash\ D)$ is ind 1-shifted Poisson, and the map $\operatorname{Bun}_G(C) \to \operatorname{Bun}_G(C \ \backslash\ D)$ is ind 1-shifted coisotropic, witnessed by the 2-shifted Lagrangian map $BL^+G^k \to BLG^k$.
3. The Lagrangian intersection $\operatorname{Bun}_G(\mathbb{B})$ is Tate 1-shifted symplectic, and the map $B\Lambda_i \to \operatorname{Bun}_G(\mathbb{B})$ associated to the inclusion of the stabilizer of a closed point is 1-shifted Lagrangian.

Consequently, the moduli stack $q\text{-Conn}_G(C,D)$ is ind 0-shifted Poisson and $q\text{-Conn}_G(C,D,\omega^\vee)$ is 0-shifted symplectic. This is the formal Tate-stack version of the shifted-symplectic and Lagrangian claims preceding it; the source presents it as a conjectural formulation because the stacks involved are generally not Artin.