Explicit triangular limit shape for Young diagrams on the diamond

Let $\Omega$ be the open diamond

$$|x|+|y|<1/\sqrt2$$

and let $\Phi$ be the limiting variational function for a homogeneous Poisson point process on $\Omega$; its derivative $\Phi'$ is the corresponding Young-diagram limit shape. Triangular limit-shape formula.

$$\Phi'(r)=\begin{cases}\sqrt2-r & \text{if $0\le r\le\sqrt2$,}\\0 & \text{if $r>\sqrt2$.}\end{cases}$$

This is the explicit formulation of the triangular shape suggested by computer simulations. It remains open in the source and, if proved, would provide the claimed explicit limit surfaces and a new derivation of the Logan–Shepp–Vershik–Kerov limit shape.