Hi Sara,
I created a new file for you (test-loop-modified2.html) to work-around the issue.
Basically, you can modify your file by following steps
1) Zoom in the timeline
2) Move the first keyframes (at 0) to 10ms for animated layers. To do so, command select all the first keyframes and move all together to 10ms.
3) Move the playhead back to 0
4) Insert keyframes at 0 for all the animated layers, but the animated properties should match the last keyframe of each layer. For example, gwd-image_6's last keyframe has opacity:0, so need to add opacity:0 at the inserted first keyframe.
5) For each animated layer, add the easing option "step-start" from 0-1 keyframe. Please use the timeline scale option so that it is easy to add easing option between keyframes.
Please let me know if this work-around works for your case on Safari.
Thanks,