Is LR(1) closed under union?

Question

Suppose I have two LR(1) languages $L_1$, $L_2$. Is $L_1 \cup L_2$ also LR(1)? References to proofs would be very helpful.

Answer 1

A simple example is $$L_1 = \{a^i b^i c^j \mid i,j\ge 0\}$$ $$L_2 = \{a^i b^j c^j \mid i,j\ge 0\}$$

Clearly, both languages are $LR(1)$. (Indeed, they are $LL(1)$.) But their union is inherently ambiguous, so not $LR(k)$ for any $k$.

This is also the usual example for non-closure of context-free languages over intersection. $L_1 \cap L_2$ is $\{a^i b^i c^i \mid i\ge 0\}$, the classic non-context-free language.