In my limited experience seeing these cases go by, no.
It's usually hard to convict these child porn cases unless you can demonstrate that the perpetrator action's were knowing and willful. Yes, some of the laws aren't like that and are strict liability, which sucks. Yes, some unwise prosecutors indict on absolutely ridiculous cases, and that sucks. But in general, if you're going to actually get a conviction in court, you really need to be able to demonstrate that the guy did it knowingly and willfully.
Even then, if your evidence of intent is too deeply technical, you conviction is at risk, because a jury absolutely hates any deep technical discussions (they are not, in general, technically-minded people). So Web browser data, for example, sucks. If you find CP images in a browser cache, then you've got to demonstrate that they got there by willful action and not by mistake. (After all, both the forensic investigator and the defense know full well that you can get porn in your browser cache with one accidental misclick.) So you've got to connect Web browser history (which used to be shorter-lived than cache entries) to the CP, which is somewhat technically complicated, and as mentioned, technical explanations are looked down on. It's worse if you find CP in unallocated space on a hard drive -- now you've really got your work cut out for you. But, I digress.
Fortunately for the prosecutor, the gross majority of people they catch make it easy. They take zero of the half-assed paranoid steps that any armchair expert on Slashdot will tell you to follow. No encryption, no "download and secure erase" policy, etc. No, they download, organize, and label hundreds of gigabytes of child porn.
Anyway, in practice, mens rea really is necessary to get a conviction. Which means one of two things here: either the prosecutor in this case is looking to make headlines and is making a bad decision (namely, they'll get their headlines but not a conviction); or, more likely, the host has knowingly harbored CP -- perhaps even specifically sought out this business, has chosen to do nothing about it, and there is substantial evidence to demonstrate this. (I think the latter is more likely not because of my faith in prosecutors, but rather because businesses providing "secure storage" but explicitly and knowingly catering to this kind of business abound.)