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I know of no Turkish expert but Jindřich Roháček is the most recent author who has been working in this area (in addition to Anatole Shatalkin of course)
Roháček J., Barták M. & Preisler J. 2016. New records of Psilidae, Piophilidae, Lauxaniidae, Cremifaniidae and Sphaeroceridae (Diptera) from the Czech Republic and Slovakia. Acta Musei Silesiae, Sci. Nat. 65: 51–62.
He might be interested, apologies to him if I've been too presumptive.
Liverpool Museum used to take frequent entomological trips to Turkey and I know they operated malaise traps there, it's a fair bet they have material.