Reformulating the higher-dimensional Kakeya conjecture via homological, algebraic-geometric, and group-theoretic frameworks

Let $E \subset \mathbb{R}^n$ be a Besicovitch (Kakeya) set, i.e., a compact set containing a unit line segment in every direction $e \in \mathbb{S}^{n-1}$. The Kakeya conjecture asserts that $\dim_{\text{H}}(E) = \dim_{\text{M}}(E) = n$ for all $n \ge 4$. Given the geometric obstructions in $n \ge 4$, I am looking for alternative structural reformulations.

1. Algebraic Topology & Homological FrameworkLet $\mathcal{G}(1, n)$ be the Grassmannian of lines in $\mathbb{R}^n$, and let $\Sigma \subset \mathcal{G}(1, n)$ be a choice of lines realizing $E$. Consider the evaluation map:$$\text{ev}: \Sigma \times [0,1] \to \mathbb{R}^n, \quad (L, t) \mapsto L(t)$$where $\text{ev}(\Sigma \times) = E$. * Can the geometric overlap of $\delta$-tubes $T_L$ around $L \in \Sigma$ be modeled via the persistent homology $H_*(\mathcal{N}(\mathcal{T}_\delta); \mathbb{Z})$ of the nerve of the tube cover $\mathcal{T}_\delta$?* Is there a known obstruction in the intersection cohomology $IH^*(E)$ or the Borel-Moore homology $H_*^{\text{BM}}(E)$ that forces $\dim_{\text{H}}(E) = n$?
2. Algebraic Geometry & Variational FrameworkFollowing Dvir's result over $\mathbb{F}_q^n$, let $\mathcal{I}(E) \subset \mathbb{R}[x_1, \dots, x_n]$ be the ideal of polynomials vanishing on $E$. For a fixed degree $d$, let $V_d = \{ P \in \mathbb{R}[x_1, \dots, x_n] : \deg(P) \le d \}$.* Let $J_e(P)$ be the directional derivative of $P$ along $e \in \mathbb{S}^{n-1}$. Since $E$ contains unit segments in all directions, $P \in \mathcal{I}(E) \implies \exists x \in E$ such that $J_e^k P(x) = 0$ for $k \le \deg(P)$. * Can the dimension of the coordinate ring $\mathbb{R}[x_1, \dots, x_n]/\sqrt{\mathcal{I}(E)}$ be bounded from below by analyzing the valuation spectrum or the Hilbert-Samuel multiplicity of $\mathcal{I}(E)$ under the directional constraint map $\mathbb{S}^{n-1} \to \mathcal{G}(1,n)$?
3. Geometric Group Theory & Homogeneous SpacesThe group of rigid motions $G = \text{SE}(n) = \mathbb{R}^n \rtimes \text{SO}(n)$ acts transitively on $\mathcal{G}(1,n)$. Let $H \cong \mathbb{R} \times \text{SO}(n-1)$ be the stabilizer of a reference line $L_0$. The space of lines is identified with $G/H$. * Let $\mu$ be a probability measure on $G$ supported on transformations that map $L_0$ to the segments in $E$. The Kakeya set is the projection of the support of $\mu$ onto $G/(\text{SO}(n))$.* Can the Kakeya maximal function estimate $\| f_\delta^* \|_p \le C_\epsilon \delta^{-\epsilon} \|f\|_p$ be reformulated as a convolution bound $\| f * \nu_\delta \|_p$ over the Lie algebra $\mathfrak{se}(n)$, utilizing the growth rate of non-abelian subgroups or random walks on $G/H$?Are there existing papers or structural obstructions regarding these specific reformulations for $n \ge 4$?