As a German, I'm pretty sure they're right. Look into glottal stop, it's not really a long pause or anything. Think of the difference of the connection of "Spiegel" and "ei" in Spiegelei, and "Schreiner" and "ei" in Schreinerei. It's this short contraction that stops airflow and then releases it again, and it's present at the beginning of the "ei" in Spiegelei, but not in Schreinerei.

Here's also the IPA pronunciations from Wiktionary:

Spiegelei: [ˈʃpiːɡl̩ˌʔaɪ̯]

Schreinerei: [ʃʁaɪ̯nəˈʁaɪ̯]

ʔ is the symbol for the glottal stop: https://en.m.wikipedia.org/wiki/Glottal_stop