Does anyone have any tips for creating a sweep, or a pipe, for this "Tee" on the surface, (very nearly,) of a Sphere?

The fillet command does not work, mostly. Did manage to fillet two vertexes in a similar rotated "Tee," which enabled sweep of those
two vertexes, but not any others.

(This model started on the Alibre forum.)

Maybe I'll try reblending the corners...