The filtered shifted symplectic and Lagrangian structure conjecture

Let $f:X\to Y$ be a morphism of Artin stacks equipped with an $n$-shifted Lagrangian structure $\gamma$, and let $\widehat{Y}_f$ be the formal completion of $Y$ along $f$. Let $\widetilde{Y}_f$ and $\widetilde{f}:X\times[\mathbb{A}^1/\mathbb{G}_m]\to\widetilde{Y}_f$ be the derived deformation over $[\mathbb{A}^1/\mathbb{G}_m]$ whose fibers at $1$ and $0$ are respectively the formal completion and the zero section. Write $\widehat{\omega}$ for the pullback of the $n$-shifted symplectic structure to $\widehat{Y}_f$, and $\gamma_X$ for the induced Lagrangian structure on the zero section.

Filtered shifted-structure conjecture. There exists a relative $n$-shifted symplectic structure $\widetilde{\omega}$ on $\widetilde{Y}_f$ over $[\mathbb{A}^1/\mathbb{G}_m]$ such that

$$1^*\widetilde{\omega}\cong\widehat{\omega},\qquad 0^*\widetilde{\omega}\cong\omega_X.$$

There also exists a relative Lagrangian structure $\widetilde{\gamma}$ on $\widetilde{f}$ over $[\mathbb{A}^1/\mathbb{G}_m]$ such that

$$1^*\widetilde{\gamma}\cong\gamma,\qquad 0^*\widetilde{\gamma}\cong\gamma_X.$$

The conjecture would make the deformation from the formal neighborhood of the Lagrangian morphism to the shifted cotangent model compatible with shifted symplectic and Lagrangian structures. The source gives no resolution.