Shifted Lagrangian thickenings of shifted Poisson derived schemes

We prove that the space of shifted Poisson structures on a derived scheme $X$ locally of finite presentation is equivalent to the space of shifted Lagrangian thickenings out $X$, solving a conjecture in shifted Poisson geometry. As a corollary, we show that for $M$ a compact oriented $d$-dimensional manifold and an $n$-shifted Poisson structure on $X$, the mapping stack $\mathrm{M}\mathrm{a}\mathrm{p}(M,X)$ has an $(n-d)$-shifted Poisson structure. It extends a known theorem for shifted symplectic structures to shifted Poisson structures.