Full Sobolev invertibility range for Hausdorff-measure screen operators

Let $n \in \{1, 2\}$, $n - 1 < d \le n$, and let $\Gamma \subset \mathbb{R}^n$ be a nonempty compact $d$-set: for some $c_1, c_2 > 0$, $$c_1 r^d \le \mathcal{H}^d(\Gamma \cap B_r(x)) \le c_2 r^d \qquad (x \in \Gamma,\ 0 < r \le 1).$$ Regard $\Gamma \times \{0\}$ as a sound-soft screen in $\mathbb{R}^{n+1}$, put $t_d = (d - n + 1)/2$ and fix a wavenumber $k > 0$. For $s > 0$ let $\mathbb{H}^s(\Gamma)$ be the space of traces on $\Gamma$ of $H^{s + (n-d)/2}(\mathbb{R}^n)$, with the quotient norm; let $\mathbb{H}^0(\Gamma) = L^2(\Gamma, \mathcal{H}^d)$ and $\mathbb{H}^{-s}(\Gamma) = (\mathbb{H}^s(\Gamma))^*$, with $L^2$ as the duality pivot. The Hausdorff-measure single-layer operator is given for bounded densities by $$(\mathbb{S}_k \varphi)(x) = \int_\Gamma \Phi_k\bigl((x, 0), (y, 0)\bigr)\, \varphi(y)\, d\mathcal{H}^d(y),$$ where $\Phi_k(X, Y) = \tfrac{i}{4} H_0^{(1)}(k|X - Y|)$ in $\mathbb{R}^2$ and $\Phi_k(X, Y) = e^{ik|X - Y|}/(4\pi |X - Y|)$ in $\mathbb{R}^3$, and is extended by continuity between these spaces. Caetano, Chandler-Wilde, Gibbs, Hewett and Moiola proved that $\mathbb{S}_k : \mathbb{H}^{t - t_d}(\Gamma) \to \mathbb{H}^{t + t_d}(\Gamma)$ is bounded for $|t| < t_d$ and invertible for $t = 0$, and that it is invertible for $|t| < \epsilon$, for some $0 < \epsilon \le t_d$, when $\Gamma$ is the attractor of an iterated function system of contracting similarities whose similar copies of $\Gamma$ are pairwise disjoint. When $\Gamma$ is the closure of a bounded Lipschitz domain in $\mathbb{R}^n$, so that $d = n$, invertibility for every $|t| < 1/2$ follows from known results on the single-layer operator.

Problem. Is $$\mathbb{S}_k : \mathbb{H}^{t - t_d}(\Gamma) \longrightarrow \mathbb{H}^{t + t_d}(\Gamma)$$ a bounded linear isomorphism for every such screen $\Gamma$ and every $|t| < t_d$? No self-similarity or separation assumption is imposed on the $d$-set.