PCR-K\"ahler equivalent metrics in the Siegel domain

Let $\mathfrak{H}$ be the Heisenberg group. From the standard CR structure $\mathcal{H}$ of $\mathfrak{H}$ we construct the complex hyperbolic structure of the Siegel domain. Additionally, using the same minimal data for $\mathfrak{H}$, that is, its Sasakian structure, we provide the Siegel domain with yet another K\"ahler structure: this structure is of unbounded negative sectional curvature, and its complex structure does not commute with the standard complex structure. However, we show that those two K\"ahler structures are {\rm PCR} K\"ahler equivalent, that is to say, essentially the same when restricted to $\mathcal{H}$.