The hosting company doesn't have to take down the content. Rather, if they don't take down the content and it is later found infringing, then the hosting company can't claim the protection of the DMCA safe harbor.
But the allegedly infringing content in the DMCA takedown in this case is (nonsensically) a trademark, so the safe harbor wouldn't protect SoftLayer against anything.
It's not functionally the same thing, because there is nothing here for them to be liable for.
"entirety of SoftLayer losing the protection"? What are you talking about?