In his book Vox Graeca, W. Sidney Allen writes (1968: 24f):
Note on ??, ??
These combinations call for some comment in view of suggestions that they do not mean what they appear to mean, i.e. a succession of two aspirated plosives. […] The reason given for doubting the straightforward interpretation of these groups is that it would be impossible to pronounce an aspirated plosive when followed by another plosive […]. This a priori dogma, frequently repeated in older works and even in some reputable modern ones, has no basis whatever in reality. Any phonetician will confirm and demonstrate the possibility of such sequences, and one can hear them as a normal feature of a number of living languages — as e.g. Armenian a?ot‘k‘ [a?othkh] ‘prayer’, or Georgian p‘k‘vili ‘flour’, t‘it‘k‘mis ‘almost’, or Abaza (N.W. Caucasian) ap‘q‘a ‘in front’. In fact there is a rule in Georgian that if a plosive consonant is followed by another located further back in the mouth, it must have the same kind of articulation as the following consonant — thus, if the second is aspirated, so must the first be (otherwise dissimilar groups can occur, as e.g. t‘bilisi ‘Tiflis’ with voiceless aspirated followed by voiced unaspirated plosive) […].
There is thus no phonetic improbability whatever about the first consonant of the groups ?? and ?? being aspirated as well as the second.
I’m afraid Sidney Allen has got his Georgian data slightly wrong. Here is what Aronson has to say about harmonic clusters (1989: 15ff):
[Describing the voiced stops:] Extremely common are the so-called harmonic clusters consisting here of a prevelar stop followed immediately by [?] and with only one release for the whole cluster: bg (??), dg (??), jg (??), ?g (??).
[Describing the aspirated stops:] The harmonic clusters here consist of a prevelar with following [k‘] and with only one release: pk (??), tk (??), ck (??), ?k (??).
Harmonic clusters will have only one release, while nonharmonic clusters will have more than one release; cf. bgera (?????) ‘sound’, where there is only one release for the harmonic cluster bg, and Tbilisi (???????) ‘Tbilisi’ (Tiflis, capital of Georgia), which has an aspirated release for the initial t and a voiced released for the b: [t‘b?l?s?].
If we symbolise the extra release with a superscript schwa (?), Tbilisi is pronounced /t‘?bilisi/ (continuing Sidney Allen and Aronson’s use of /‘/ for aspiration). Because aspiration is a feature of the release, harmonic clusters have only one instance of articulation. Sidney Allen’s p‘k‘vili () ‘flour’ and t‘it‘k‘mis () ‘almost’ are therefore pronounced [pk‘vili] and [t‘itk‘mis] without the double aspirations that he postulated. Please note that the [p] in the former and the [t] in the former are still clearly /p‘/ and /t‘/ for two reasons: Firstly, they form part of harmonic clusters, so their aspiration can be deduced; secondly, Georgian has no unaspirated stops (*[p] and *[t]), only voiced ([b] and [d]), voiceless aspirated ([p‘] and [t‘]) and voiceless glottalised ([p’] and [t’]), so the loss of aspiration does not make these sounds merge with others.
My point is that ?? and ?? are most likely either to have been pronounced as harmonic stops ([pt‘] and [kt‘]) or as nonharmonic stops ([p‘?t‘] and [k‘?t‘]); Sidney Allen’s suggestion that the likely pronunciation must have been [p‘t‘] and [k‘t‘] with double aspiration and one release seems less probable based on the evidence of Georgian.