The Vank of Surp Astvatsatsin at Lusahovit, Armenia
It was first built in the 5th or 6th century and its roof was restored in 1213.
In Armenia, conical or pyramidal roofs were equipped with batten seams.

Cross section of the Vank of Surp Astvatsatsin at Lusahovit
(From "Early Christian Achitecture of Armenia" by Murad Hasratian, 2000)
A conical roof was built over the domical ceiling across a vacant space,
which would have commonly been filled with ballast or stones.
Looking such a cross section leads to an idea that
the origin of conical roofs would have been wooden roofs
over stone buildings to protect them from rain. This reminds one
of sedge-hat-type temples in the Himachal region, India.

A Hindu Temple in Bhalmaur, Himachal Pradesh, India
To protect stone temples (the style of which was brought
from the plain of lower India) from Himalayan high rainfall,
wooden roofs like sedge-hats were built over them.