Milnor's conjecture on rational-angle Lobachevsky relations

For an angle $\theta$, let $\operatorname{L}(\theta)$ denote the Lobachevsky function. Consider angles $\theta$ that are rational multiples of $\pi$, and rational coefficients $q_1,\ldots,q_n$. Milnor's conjecture. Every $\mathbb{Q}$-linear relation

$$q_1\operatorname{L}(\theta_1)+\cdots+q_n\operatorname{L}(\theta_n)=0$$

is a consequence of

$$\operatorname{L}(\pi+\theta)=\operatorname{L}(\theta),\qquad \operatorname{L}(-\theta)=-\operatorname{L}(\theta),$$

and

$$\operatorname{L}(n\theta)=n\sum_{k\!\!\!\!\mod n}\operatorname{L}(\theta+k\pi/n).$$

This conjecture concerns the linear relations underlying hyperbolic volume and scissors congruence. The source attributes it to Milnor and relates it to the sufficiency of the Dehn invariant conjecture for $\mathbb{H}^3$; no resolution is given.