3-extendability conjecture for optimal 1-embedded graphs on the Klein bottle

Let $G$ be an $8$-regular optimal $1$-embedded graph on the Klein bottle. A graph is $3$-extendable if every matching of size $3$ extends to a perfect matching.

3-extendability conjecture. Every $8$-regular optimal $1$-embedded graph $G$ on the Klein bottle is $3$-extendable.

This conjecture concerns matching extendability and follows discussion of the classification of $4$-regular quadrangulations on the Klein bottle. The status of the claim is not resolved in the supplied source.