This page links to 8 experimental triply periodic surfaces of genus 6. They can be obtained from minimal heptagons that have straight horizontal edges at the top and bottom, and in between segments in symmetry planes parallel to the coordinate axis. There are 12 combinatorially possible cases:

Cases I/II distinguish whether the three corners in the vertical symmetry planes lie in one component (I) or in both components (II);

Cases A/B distinguish where the Gauss map points up along the horizontal lines;

Cases 1/2/3 distinguish where the Gauss map points up along the planar symmetry curves.

More details can be found in the divisor images and/or the Mathematica notebooks.

I have been able to close the periods numerically to my satisfaction in the 8 cases below.

Existence/Non-Existence will be difficult to establish on a case-by-case basis, as the period problems are 3-dimensional (not taking the adjustment of the López-Ros factor into account). The reason for these experiments is to learn what limits can occur, and what new techniques will be needed to obtain these surfaces from their limits using gluing constructions.