Łaba–Wang conjecture on spectral self-similar measures

Let $0<\rho<1$, let $\mathcal{D}=\{0,d_1,\ldots,d_{m-1}\}$ be a digit set, and let ${\bf p}=(p_0,\ldots,p_{m-1})$ be a probability vector. For the iterated function system $\phi_j(x)=\rho(x+d_j)$, let $\mu_{\rho,\mathcal{D},{\bf p}}$ be the unique probability measure satisfying

$$\mu(E)=\sum_{j=0}^{m-1}p_j\mu(\phi_j^{-1}(E)).$$

Łaba–Wang conjecture. If $\mu_{\rho,\mathcal{D},{\bf p}}$ is a spectral measure, then ${\bf p}$ is equal-weight, so $p_j=1/m$; $\rho=1/N$ for some integer $N$; and there exists $\alpha>0$ such that $\mathcal{D}=\alpha\mathcal{C}$, where $\mathcal{C}\subset\mathbb{Z}^+$ tiles $\mathbb{Z}_N$, meaning that there is a set $\mathcal{B}$ with $\mathcal{C}\oplus\mathcal{B}\equiv\mathbb{Z}_N\pmod N$. This conjecture seeks a classification of spectral self-similar measures by forcing equal weights, reciprocal-integer contraction ratio, and a digit set arising from a cyclic tiling structure; the source does not provide evidence that it has been resolved.