How strict is Lufthansa about pet carrier dimensions?

Since this answer hasn't seen much love here, I'm cross-posting answers from the linked FlyerTalk forum post made by OP.

While not an answer specific to Lufthansa, one poster states:

Most carriers are extremely lenient with soft side carriers as they can be squeezed in spaces that a hard sided carrier will not fit in.

In fact AA even mentions this fact on their site where they say, "The maximum size for cabin pet carriers is 19" long x 13" wide x 9" high. Soft-sided pet carriers such as Sherpa bags may exceed these dimensions slightly because they are collapsible."

However, Lufthansa is part of Star Alliance while American Airlines is part of oneworld, and they are different airlines anyway so policies could differ.

Absolutely strict or we got totally ripped off. It may have everything to do with the fact that we were taking our dog out of Istanbul airport to Germany and they made us buy one of their crates from cargo otherwise they would not allow my dog on the flight. They said they only had a large crate left and so ended up charging us in total an extra 360 euros which was almost the cost of my flight ticket. Turkish staff were really horrible.