Keller's invariant-measure representation conjecture for regular Toeplitz $free$ systems

Let $\mathscr{B}subseteqmathbb{N}$ be such that $\eta^*$ is a regular Toeplitz sequence. Let $H$ be the associated dynamical system with transformation $R$, let $\varphi,\underline{\varphi}$ define the endpoints of the fiber interval $[\underline{\varphi},\varphi]$, and let $X_\varphi$ be the corresponding $\mathscr{B}$-free subshift. For $hin H$ and $xin\ta{0,1\ta}^{\mathbb{Z}}$, define

$$M_H(h,x)=\underline{\varphi}(h)+xcdot(\varphi(h)-\underline{\varphi}(h)).$$

Keller's conjecture. For every $nuin\ta{P}(X_\varphi)$, there exists $rhoin\ta{P}(Hxtimes\ta{0,1\ta}^{\mathbb{Z}},Rxtimessigma)$ such that, for every measurable $Asubseteq X_\varphi$,

$$nu(A)=\int_{Hxtimes\ta{0,1\ta}^{\mathbb{Z}}}\mathbf{1}_A(\underline{\varphi}(h)+xcdot(\varphi(h)-\underline{\varphi}(h)))\,drho(h,x).$$

Equivalently, $nu=(M_H)_*\rho$ for some such $\rho$. The conjecture gives a complete representation of invariant measures on $X_\varphi$ through measures on the skew-product system; its resolution is not established by the supplied source context.