Was mtDNA haplogroup H present in Mesolithic Iberia?

Today, the predominant mtDNA haplogroup in Europe is H.  It is absent from numerous ancient DNA samples everywhere to the north of the Olive Oil-Butter line in Europe before the advent of farming in Europe, and remains rare until the Copper Age that coincides with the advent of the Corded Ware culture in Central and Eastern Europe, and with the Bell Beaker culture of Western Europe.

There are a couple of studies, however, that claim to have found mtDNA haplogroup H in Iberia in the archaeological period immediately prior to the arrival of farming and herding there, sometimes called the Mesolithic era or Epipaleolithic era, that corresponds to the pre-Neolithic Holocene era.  There are quite early, but not as definitively Mesolithic, examples of mtDNA H in Italy and Southeastern Europe.  There is also mtDNA H in the early Neolithic Fertile Crescent (which is earlier in absolute dating than the Neolithic elsewhere).

The Iberian studies are the strongest evidence far mtDNA H having an origin in the Franco-Cantabrian refugia, as opposed to arising with the first farmers (and herders) or subsequent migrations that had run their course by the end of the Bronze Age in Europe (i.e. prior to 1200 BCE), or earlier.

Bell Beaker blogger challenges the validity of these studies in a pair of recent posts here and here.  He focuses on two points:

1.  The context of the ancient mtDNA H samples called Mesolithic is dubious, coming from old excavations of shell middens where samples were interspersed with Neolithic era pottery as well as older materials, and is also subject to other concerns related to its provenance.

2.  C14 dating of bone from people who had a shellfish heavy diet as the people buried in the shell middens that were a source for key ancient DNA samples are not reliable in the way that they are for terrestrial individuals.  Basically, their diet altered the baseline C14 levels upon which C14 data is premised from the standard assumptions concerning that baseline that are used to date the bones.

These criticisms, at a minimum, are not frivolous.  And, if these troublesome data points are excluded, in part on the intuition that extraordinary claims should require extraordinary evidence, then the rest of the ancient DNA data from Europe falls together into a much simpler picture.  In that picture, a much larger share of modern Europe's maternal genetic heritage arrived with the first farmers (or second wave of farmers), rather than dating back to integration of the continent's indigenous hunter-gatherer population into frontier farming communities (a more plausible possibility in the case of women than in men).  Bell Beaker blogger, in particular, doubts that mtDNA H could really have been so geographically confined for so long if it was really indigenous.

On the other hand, new autosomal DNA evidence seems to indicate that modern Europeans have fairly substantial genetic European hunter-gatherer ancestry.

I'm interested in hearing the opinions of others on these methodological critiques of this key evidence, upon which key conclusions about European prehistory and population genetics rests.