Closed-form conjecture for the rectangular optimal shrinker

Let $\Upsilon_{\gamma,\beta}(y)$ be the shrinker defined in equation (Upsilon), and let $\mathsf{z}^{+}_{\gamma,\beta}$ and $\mathsf{z}^{-}_{\gamma,\beta}$ be the associated spectral-edge quantities. Closed-form conjecture for the rectangular optimal shrinker. For all $\gamma\in (0,\infty)$ and $\beta\in (0,1)$ with $\gamma\beta<1$, the shrinker $\Upsilon_{\gamma,\beta}(\cdot)$ has the closed-form expression

$$\Upsilon_{\gamma,\beta}(y) = \begin{cases} \frac{1}{y}\sqrt{(y^2-\mathsf{z}^+_{\gamma,\beta})(y^2-\mathsf{z}^-_{\gamma,\beta})}&\quad\textrm{if } y>\sqrt{\mathsf{z}^+_{\gamma,\beta}},\\ 0 &\quad\textrm{if } y\le \sqrt{\mathsf{z}^+_{\gamma,\beta}} \end{cases}.$$

This would extend the known explicit optimal shrinker for the full SVD to the rectangular setting; the source reports exhaustive numerical verification, but does not provide a proof.