Purity or rather pure breeding of the Akhal Teke is currently one of the most debated and debatable aspects of practical Akhal Teke breeding.
After several millennia of classic pure landrace breeding with little to no influences from horsebreeds not originating from within the Turkoman pool of horse breeds, the two last officially bred Turkoman breeds - Akhal Teke and Yamoud - were crossbred, mainly with English Thoroughbreds, in a major way as per Russian/Soviet order during the 20th century.
Part of the reason for ongoing discussion are the extremely different positions of eastern (Russian/Soviet) and western (worldwide) breeders on purebreeding, the diverging habits and methods of pure breeding in both cultural spheres and the fact that the infusions were politically caused and are now breed-genetically defended. In fact, up to the 80ies the western buyers were rarely aware of the impurity of some of the horses sold to them and the breed in general.
In horse breeding, as in most animal breeding these days, purity of the gene pool is of actual (in some cases) and purported import, with the main competitor of the Akhal Teke, the Arabian, claiming absolute purity as a marketing and authentic value. Breeders of so-called pure breeds worldwide have a set expectation of what pure breeding is, how it is effected in very old and how in newer breeds, how it is maintained and what is logical in these respects.
Unfortunately the Akhal Teke breed breaks these expectations on the whole spread of them. Not just that it is an ancient breed which ought to have been kept pure right from the start of being managed by the Russian/Soviet military, the infusion of foreign blood was mishandled in the gravest manner (if you actually believe that it was well-meant), the subsequent closure of the studbook was seriously botched, including highly impure horses, excluding a quite large number of purer ones, and it was effected in a manner not breed-scientifically acceptable. Then, as per this closure, an artificial seal of purity was applied to the breed, misleading newcomers to the breed regarding the actual facts. As a result, especially among western breeders, the problem was exacerbated up to the current status quo of having no truly purebred Akhal Teke left among the so-called pure stock of the closed studbook.
Additionally many dedicated people suffered often quite extreme financial losses upon learning that their bought-as-purebred horses, at prices of often better than 40,000 USD, have been evicted from the closed studbook and now actually value less than 4,000 USD. This even though in-depth checking of the impurity of said horses often was much less than that of horses inside the closed studbook and even though some of these horses were of rare and important lineage not anymore found within the closed studbook.
The situation is ongoing and unresolved.
The breed was not treated and is not treated in a manner which is expected of a purebred breed, no efforts are made to ensure this, and an important part of the gene pool is still left out of the main one, inspite of dire need to at least ensure a better diversity against future inbreeding depression and genetic drift. Additionally owners of these artifically evicted horses are left standing in the financial rain of being unable to properly market these horses as what they are: just the exact same kind of Akhal Teke which are inside the closed studbook. Genetically there is absolutely no difference between a horse which has an English Thoroughbred ancestor in 6th generation dating 1934 and a horse which has an English Thoroughbred ancestor in 6th generation dating 1960.
To better understand all this, it is well worth reading up the sections Purity and Studbook, including in-depth looks at the pedigrees shown as samples.