This paper addresses a computational philology task focused on the automatic restoration of textual gaps (i.e., lacunae) in the Herculaneum Papyri, whose Ancient Greek texts are inherently fragmentary due to damage caused by carbonization. The objective of this work is to show the preliminary results concerning the development of a web-based suggestion service for proposing plausible supplements to fill lacunae, thereby supporting the philological process of producing new critical editions within a new web-based digital scholarly editing environment. To automatically provide such suggestions, we have developed systems that generate textual supplements in Ancient Greek, employing both neural (BERT-like) and statistical (n-gram) language modeling approaches.