Broderick–Fishman–Reich conjecture for intrinsic approximation on missing-digit fractals

Let $k\in\mathbb{N}$, and let $\mathcal{K}$ be a proper missing-digit fractal in $[0,1]^k$ that is not contained in any proper affine subspace. Let $\mu$ be its Cantor–Lebesgue measure, and let $W_k^{\mathcal{K}}(\tau)$ denote the set of points intrinsically approximable at rate $\tau$ by rational vectors lying in $\mathcal{K}$. Broderick–Fishman–Reich conjecture. For all $\tau>0$,

$$\mu\bigl(W_k^{\mathcal{K}}(\tau)\bigr)=0.$$

This conjecture concerns intrinsic Diophantine approximation on fractals. The supplied context records related zero-measure results for $\tau>1$ in dimension one and $\tau>1/k$ under affine irreducibility, but does not resolve the stated all-$\tau>0$ assertion.