Compatibility conjecture for the shifted symplectic structures on de Rham mapping stacks

Let $C$ be a smooth and proper curve, let $(X,\omega)$ be an $n$-shifted symplectic stack, and let $\int_C\mathrm{ev}^*\omega$ be the induced closed one-form on $\mathrm{Map}(C,X)$. The proposition gives an isomorphism

$$\mathrm{Map}(C_{\mathrm{dR}},X)\cong \mathrm{T}^*_{\int_C\mathrm{ev}^*\omega}[n-2]\mathrm{Map}(C,X).$$

Compatibility conjecture. This isomorphism is compatible with the $(n-2)$-shifted symplectic structures on both sides. The mapping stack $\mathrm{Map}(C_{\mathrm{dR}},X)$ is known to carry an $(n-2)$-shifted symplectic structure, while the compatibility asserted here is not resolved in the supplied text.