$\varphi$-rationality of perfect Schubert schemes

Let $\Fl_{\calG}$ be an affine flag variety, and let $\Fl_{\calG,\leq w}$ be its Schubert subvariety indexed by a double coset $w$ of the Iwahori--Weyl group. Its perfection is denoted by $\Fl_{\calG,\leq w}$. $\varphi$-rationality conjecture. The perfect Schubert schemes $\Fl_{\calG,\leq w}$ are $\varphi$-rational. This expectation concerns the singularities of perfect Schubert schemes and extends the $\varphi$-rationality phenomena defined for perfect characteristic-$p$ local rings; the source does not provide evidence resolving it.