Canonical identification of minimal and deformed maximal extensions

Let $i$ be the inclusion of the relevant $K$-orbit, let $\phi$ be the corresponding local system, and let $i_!\phi$ and $i_+\phi$ denote its minimal and maximal extensions as $(\mathcal{D}_\lambda,K)$-modules. Let $\mathcal{I}(\lambda_t,Q,\phi)$ be the family of standard Harish-Chandra sheaves, with specialization at $t=1$ denoted by $\mathcal{I}(\lambda_t,Q,\phi)|_{t=1}$. There is a canonical morphism $i_!\phi\to i_+\phi$ and a canonical morphism $\mathcal{I}(\lambda_t,Q,\phi)|_{t=1}\to i_+\phi$. Canonical extension conjecture. There is a canonical isomorphism of $(\mathcal{D}_\lambda,K)$-modules

$$i_!\phi\cong \mathcal{I}(\lambda_t,Q,\phi)|_{t=1},$$

which identifies the canonical morphism $\mathcal{I}(\lambda_t,Q,\phi)|_{t=1}\to i_+\phi$ with the canonical morphism $i_!\phi\to i_+\phi$. This would identify the minimal extension with the specialization of the deformed standard Harish-Chandra sheaf and clarify the intrinsic nature of the construction; the source provides no evidence that the conjecture has been resolved.