According to the AIM section
that Mark cited, the "DME required" statement in the notes section doesn't solely apply to the missed approach; it also can apply anywhere outside of the final approach segment. Based on the DME-defined stepdowns preceding the final approach fix, that appears to be the case on this approach.
"When radar or other equipment is required on
portions of the procedure outside the final approach
segment, including the missed approach, a note will
be charted in the notes box of the pilot briefing
portion of the approach chart (e.g., RADAR
REQUIRED or DME REQUIRED)."