Algebraic dependence conjecture for hyperelliptic constants

Let $\pi_n$ denote the hyperelliptic constant associated with the index $n$. Let $n\in\mathbb{N}$ be composite and not a prime power, so that $n\neq p^m$ for $p\in\mathbb{P}$ and $m\in\mathbb{N}_0$. Two numbers are $\overline{\mathbb{Q}}$-linearly dependent if they satisfy a nontrivial linear relation with coefficients in $\overline{\mathbb{Q}}$. Algebraic dependence conjecture. There exists a divisor $d\mid n$ with $1<d<n$ such that $\pi_n$ and $\pi_d$ are $\overline{\mathbb{Q}}$-linearly dependent. This asks which hyperelliptic constants are algebraically independent of one another; the source presents the assertion as a suspicion, and no resolution is supplied.