TITLE: Consistency strength of the failure of Shelah's Strong Hypothesis (SSH)
QUESTION [7 upvotes]: Some known facts about SSH (Shelah's Strong Hypothesis):
i) "$0^\sharp$ does not exist" implies SSH.
ii) SSH implies SCH (Singular Cardinal Hypothesis).
iii) The failure of SCH is equiconsistent with
$\exists \kappa (o(\kappa) = \kappa^{++})$.
What else is known about the consistency strength of the failure of SSH?

REPLY [5 votes]: Additional informations from

Pierre Matet. Large cardinals and covering numbers,
  Fundamenta Mathematicae 205 (2009), 45-75.
  doi:10.4064/fm205-1-3

on the first page:

And, as Moti Gitik pointed out to the author, one obtains a model of
  "$u (\omega_1 , \omega_{\omega}) > {\omega_{\omega}}^+$
  (and hence $\neg$SSH) + SCH" by adding $\aleph_{\omega + 1}$
  Cohen reals to a model of
  "for every infinite cardinal $\nu$, $2^{\nu}$ equals $\nu^{++}$
  if $\nu = \omega_{\omega}$, and $\nu^+$ otherwise".

and

It is not known whether the failure of SSH is equiconsistent with that of SCH.