Interior variant of Falconer's conjecture for pinned chain distance sets

Let $E\subset \mathbb{R}^d$ be compact with Hausdorff dimension $\dim_{\rm H}(E)>\frac{d}{2}$. For $n\in\mathbb{N}$ and $x\in E$, let $\Delta_x^n(E)$ denote the pinned $n$-chain set of $E$ based at $x$. Interior variant of Falconer's conjecture. For every $n\in\mathbb{N}$, there exists some $x\in E$ such that

$$\Delta_x^n(E)^{\circ}\neq\varnothing.$$

This strengthens Falconer's distance-set conjecture by requiring nonempty interior for every pinned chain length, rather than merely asserting a positive-dimensional or positive-measure distance set. The source provides motivation from interior results for distance sets and proves related constructions for suitable Cantor sets, but does not state that this conjecture has been resolved.