I never knew that corded ware penetrated deep into the Balkans. According to this recent paper, it certainly did, even as far as present-day Greece.
Abstract: The analysis of corded ware and accompanying artifacts reveals the nature of its appearance across the Central and Southern Balkan Eneolithic during three cultural-chronological horizons. The first horizon corresponds to the Early Eneolithic, namely the Bubanj-Salcuta-Krivodol cultural complex (BSK), while the second corresponds to the Cotofeni culture. The third horizon, showing chronological continuity with the second, and set within the Late Eneolithic/Early Bronze Age, has a site distribution that encompasses the territory of nearly the entire Balkan Peninsula, where corded ware is found together with other steppe elements which are present in large numbers, such are burials under mounds and the appearance of the domestic horse.
Aleksandar Bulatovic, Corded Ware in the Central and Southern Balkans: A Consequence of Cultural Interaction or an Indication of Ethnic Change?, The Journal of Indo-European Studies, Volume 42, Number 1 & 2, Spring/Summer 2014
Population genomics of Early Bronze Age Europe in three simple graphs