A constant regression characterization of a Marchenko-Pastur law

Lukacs type characterization of Marchenko–Pastur distribution in free probability is studied here. We prove that for free $\mathbb{X}$ and $\mathbb{Y}$ when conditional moments of order $1$ and $-1$ of $(\mathbb{X}+\mathbb{Y}{)}^{-1\mathrm{/}2}\mathbb{X}(\mathbb{X}+\mathbb{Y}{)}^{-1\mathrm{/}2}$ given $\mathbb{X}+\mathbb{Y}$ are constant then $\mathbb{X}$ and $\mathbb{Y}$ have Marchenko–Pastur distribution.