In the same way as Fusheng Wei’s doubly periodic genus 2 surface can be viewed as a limit of triply periodic genus 4 surfaces, the higher genus Rossman-Thayer-Wohlgemuth surfaces are limits of triply periodic surfaces. Here is one of example of genus 6.
The surfaces can also be obtained by reflecting minimal nonagons that have edges (resp. vertices) on the faces (resp edges) of a box. In the divisor notation explained in the genus 5 box type series, this would be a surface of type (++–|+). There are certainly many more like these.
While the doubly periodic RTW surfaces come in a 1-parameter family (insisting on reflectional symmetries), the triply periodic versions form a 2-dimensional family, asking the scintillating question what the other limits will look like. Some of them are shown above leaving room for speculation and investigation.