Hikita-Iyama isomorphism conjecture for the symmetric group center

Let $S_n$ be the symmetric group, let $Z(\mathbb{C}S_n)$ be its center, and let $\Theta\colon Z(\mathbb{C}S_n)\xrightarrow{\sim}\mathbb{C}[\mathcal{C}(n)]^{\mathbb{T}}$ be the isomorphism constructed in the paper. Equip both sides with their natural filtrations, and write $\operatorname{gr}\Theta$ for the induced graded isomorphism. Hikita-Iyama conjecture. The isomorphism

$$\operatorname{gr}\Theta\colon \operatorname{gr}Z(\mathbb{C}S_n)\xrightarrow{\sim}\mathbb{C}[(S^n(\mathbb{A}^2))^{\mathbb{T}}]$$

coincides with the isomorphism constructed in Section 2 of the cited work. This conjecture concerns compatibility between the filtered center of the symmetric-group algebra and the geometry of the symmetric product; its resolution is not established in the supplied text.