Let $k=\bar{k}$ be an algebraically closed field [Thanks to Mohan for pointing out that this is a necessary assumption in the comments!].
Let $A=k[x_{1},\ldots,x_{n}]$ and $I=(f_{1},\ldots,f_{r})\subseteq A$ an ideal. Let $B=k[y_{1},\ldots,y_{m}]$ and $J=(g_{1},\ldots,g_{l})\subseteq B$ an ideal. Suppose $A/I$ and $B/J$ are unique factorization domains (UFD). Is the same true for $(A/I)\otimes_{k} (B/J)$?
Attempt: thanks to Gauss' lemma we know this to be true when $I=0$ or $J=0$. So we have an UFD $R:=A\otimes_{k}B$ and two ideals $I\subseteq R$ and $J\subseteq R$ such that both $R/I$ and $R/J$ are UFDs. Moreover, these $2$ ideals come from $A$ and $B$ respectively, so they "do not interact too much with each other". This vague condition, whatever it means, is certainly a necessary condition for the claim to be true, since otherwise we could just intersect a conic in the plane with a line tangent to it to get a counterexample: $$ R=k[x,y], I=(x), J=(x-y^{2}). $$ But I was not able to make this vague condition precise in algebraic terms, and I am not even sure if I should expect the result to be true or not.