The minimal triple point number of the $2$-knot that is the 2-twist spin of the knot $5_2$ is bounded between 8 and 12. The movie suggested by Fox’s example 15 has triple point number 12. To improve this bound we follow Shin Satoh’s work on triple point number and construct virtual surfaces that have small triple point number and for which Mochizuki’s 3-cocycle vanishes. This is joint work with Seonmi Choi, Hongdae Kim, Sangsu Lee, and Seong Yeop Yang.