I'm imagining [@Anindya](https://forum.azimuthproject.org/profile/1950/Anindya%20Bhattacharyya)'s counterexample like a stubby jellyfish -- a big blob at the top with some bits hanging down that are incomparable with everything except the blob.

Left-multiplication fails incredibly easily in this order. If \$$x\$$ is not an A-word, and \$$y \le y'\$$ are distinct in any way, then prepending \$$x\$$ necessarily makes \$$x \otimes y\$$ and \$$x \otimes y'\$$ incomparable. For the same reason, right-multiplication easily preserves the order: it doesn't touch the first letter, so there's no chance of it disturbing the order.