Conjecture on non-Gaussian and almost-everywhere Gaussian fluctuations for substitutions

Let $\sigma$ be a primitive substitution of constant length $d$ with eigenvalue $1$ and eigenvector $\gamma$. Write $S_{\lfloor d^n t\rfloor}(\gamma)$ for the associated ergodic sum and let $V_n=\operatorname{Var}(S_{\lfloor d^n t\rfloor}(\gamma))$. Fluctuation conjecture. (1) There exists $t\in[0,1]$ such that the limit distribution of

$$\frac{1}{\sqrt{V_n}}S_{\lfloor d^n t\rfloor}(\gamma)$$

is not Gaussian. (2) For Lebesgue almost every $t\in[0,1]$,

$$\frac{1}{\sqrt{V_n}}S_{\lfloor d^n t\rfloor}(\gamma)$$

converges in distribution to the normal law $N(0,1)$. The paper establishes Gaussian convergence for almost every $t$ only for infinitely many primitive substitutions satisfying additional conditions, while examples show that non-Gaussian limit distributions can occur; the stated claim is therefore not resolved in the supplied text.