Existence of bounded-degree Grassmann eposets

Let $q$ be a prime power, let $d\in\mathbb{N}$, and let $0<\gamma<1$. A $d$-dimensional measured poset $X$ is sparse when $|X(0)|=n$ and $|X(d)|=O(n)$, where the implied constant may depend on $q$ and $d$, but not on $n$. For $i<d$, let $\Pi_i$ denote the distribution obtained by first choosing a $(d+1)$-dimensional subspace $s_d\in X(d)$ and then, given an $i+1$-dimensional space $s_i$, choosing $s_{i-1}$ uniformly among the codimension-$1$ subspaces of $s_i$. An $(\vec r,\vec\delta,\gamma)$-eposet is understood with the parameters specified below. Existence of bounded-degree Grassmann eposets. For every prime power $q$, every $d\in\mathbb{N}$, and every $0<\gamma<1$, there exists an infinite sequence of natural numbers $n_1<n_2<n_3<\dots$ such that, for every $n=n_j$, there is a $d$-dimensional measured poset $X$ satisfying: (i) $X$ is sparse; (ii) $X$ embeds as a poset into $\operatorname{Gr}_q(n,d)$, each $\Pi_i$ is obtained by the experiment above, and $X$ is downward closed; and (iii) $X$ is an $(\vec r,\vec\delta,\gamma)$-eposet with

$$r_i=\frac{1}{q^{i+2}-1},\qquad \delta_i=1-r_i.$$

The conjecture proposes a bounded-degree analogue of high-dimensional expanders inside the Grassmann poset; its status is not established by the supplied material.