For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of the equation $x^2=x$ in $A_i$. Denote by $\mu(A_i)$ the group of roots of the unity in $A_i$, i.e., elements of finite multiplicative order in $A_i$.

Problem. Is the map $$\mu(A_1)\times\mu(A_2)\to\mu(A_1\otimes_{\mathbb Z}A_2),\;\;(u,v)\mapsto u\otimes v,$$ surjective?

(The problem is posed 9.07.2018 by Tristan Tilly from Leiden University on page 25 of Volume 2 of the Lviv Scottish Book).

Prize for solution: An eternal welcome in Netherlands, and a bottle of liquor of your choice.