Truesdell Bill, the following is an extract from Jean-Pierre Chapleau
project on screen bottom board:

> as we can read in his study, the explanation of the increased Varroa
> population with an OPEN SBB would be that the lowering of temperature
affect
> the larva maturation time, perhaps delaying 2 to 3 days the normal adult
> emerging time.The cell being closed 2 or 3 days more would allow MORE
young
> Varroa female to become mature before the adult emerge.
> Hence, the increase of Varroa population during colder temperature with an
> OPEN SBB.
> Hope I have helped.

There were some references with this extract which I forgot to mention. Here
they are:

1. Fries, Ingemar, Varroa in cold climates: Population Dynamics,
Biotechnical Control and Organic acids, in Living with varroa, 1993, pp37-48
where we can read:
 "(....)mite population seems to grow faster in cooler climates than in warm
areas(...)it has been suggested that climates factors are decisive in
determining the mite population growth although the nechanism remain
unknown."

2.Kraus,B and Velthuis, H.H. The impact of temperature gradients in the
brood nest of honeybee on reproduction of Varroa jacobsoni (abstract # 38)
in Abstracts from the second international conference on africanized honey
bees and bee mites. American Bee Journal, vol 140, no 10 (2000) p.827
Both scientists found that colonies in which brood nest temperature has been
artificially lowered have doubled their Varroa population in comparison with
a witness group. Then, laboratory test allowed them to set at 33C the
fastest reproduction temperature for Varroa.
(Kraus and Velthuis results were not published at the time J.-P.Chapleau
build up his trials protocoles in 2000, those results have been published
only in October 2000.).

On the basis of those 2 references, it is plausible to assume that an OPEN
screen BB could lower the brood nest temperature (1-2-3C? we do not know)
and consequently extend the adult emerging time, allowing Varroa to be more
numerous in such a situation.
To answer your question, yes it is purely hypothetical and has been
presented as such in Chapleau project.

Best regards,

Docteur André Simoneau, b.sc.,d.m.v.
CQIASA
Laboratoire de pathologie animale
867, boulevard L'Ange-Gardien
L'Assomption, Québec
CANADA    J5W 4M9
Tél. (450)-589-5745 poste 275
Télécopieur:   (450)-589-0648