Perfectoid approximation conjecture for smooth closed subschemes

Let $K$ be a perfectoid field with tilt $K^\flat$, let $X\subset \mathbb{P}^n_K$ be a smooth closed subscheme, and let $\tilde{X}\subset (\mathbb{P}^n_K)^\mathrm{ad}$ be a small open neighborhood of $X$. Perfectoid approximation conjecture. There exists a closed subscheme $Y\subset \mathbb{P}^n_{K^\flat}$ such that

$$Y^\mathrm{ad}\subset \pi^{-1}(\tilde{X}),\qquad \dim Y=\dim X.$$

If true, this would extend the proof of weight-monodromy beyond complete intersections by approximating the relevant fractal globally by an algebraic variety; the conjecture is presented as open.