Linear recurrence conjecture for Mahler measure iterates of algebraic units

Let $M$ denote Mahler's measure, and write $M^{(n)}$ for its $n$-fold iterate. For an algebraic unit $\alpha$, consider the sequence of real numbers $(\log(M^{(n)}(\alpha)))_{n\geq k}$, where $k$ is a constant that may depend on $\alpha$. Mahler-measure iteration conjecture. For every algebraic unit $\alpha$, there exists a constant $k$ such that the sequence

$$(\log(M^{(n)}(\alpha)))_{n\geq k}$$

satisfies a linear homogeneous recursion. The degree-four result establishes this phenomenon for algebraic units of degree $4$, while the general case remains open and is motivated by computational evidence.