Conjecture on symbolic-square generators and the square of the transformed fat ideal

Let $R=k[x,y,z]$, let $\widetilde{J}\subset R$ be the transformed fat ideal, and let $D_1,\ldots,D_s$ be the source inversion factors of the associated birational map. Let $(x,y,z)$ denote the homogeneous maximal ideal, and let $f,g$ be the forms appearing in the generators $xf,yf,xg,zg$ of $\widetilde{J}$.

Conjecture on the transformed ideal.

$$\widetilde{J}^{(2)}=(D_1,\ldots,D_s),\qquad \widetilde{J}^2=(x,y,z)\widetilde{J}^{(2)}=(xf,yf,xg,zg)\widetilde{J}.$$

These identities would give an explicit description of the second symbolic power and relate the ordinary square to it through the homogeneous maximal ideal and the displayed generators. The source does not indicate that this conjectural statement has been resolved.