Modulated Self Shadow is only ever disabled and enabled for Skeletal Meshes and does not have any effect on Static Meshes. You can use the command r.Shadow.EnableModulatedSelfShadow 1 to enable self shadowing and see that it is enabled and disabled for Skeletal Meshes.
This leaves planar objects, like the cube in the test project, with noticeable edges that are not self-shadowed.
In the attached image, the back actors are Static Meshes and the front ones are Skeletal Meshes. This scene has r.Shadows.EnableModulatedSelfShadow 0, which is default.
1. Open the attached Project
2. Set the Preview Rendering Level to anything on Mobile/HTML5
3. Use the r.Shadow.EnableModulatedSelfShadow 1 and then back to 0.
Result: Modulated Self Shadow is never disabled for static meshes
There's no existing public thread on this issue, so head over to AnswerHub just mention UE-41581 in the post.