Srimathe Ramanujaya Namaha,
Swami Vedanta Desikan has made a commentary in Rahasyatraya Saaram about Varaha Charama Slokam.
1) There is a prerequisite for BhagavAn Sriman Narayana to execute HIS promise.
The devotee must perform Saranagati (COMPLETE SURRENDER a.k.a SWATHANTRA PRAPATTI) at least ONCE in their lifetime while sound in mind and body, acknowledging the Supreme Lord as the sole means (Upayam) and goal (Upeyam) of salvation.
2) For such “completely surrendered” devotees, BhagavAn “Removes of the Antima Smrithi Burden”. General Karma-Gyana-Bhakti yoga maarga tradition followers require the dying person to remember BhagavAn in their final breath (antima smrithi), because they do “ANGA PRAPATTI” only. This type of prapatti is surrendering to BhagavAn to remove their hindrances to start their karma-gyana-bhakti yoga maargam, and it’s not COMPLETE SURRENDER (SWATHANTRA PRAPATTI)
For such Sharanagathas (Swathantra Prapatti), In old age or severe illness, pain often leaves the senses unresponsive, Supreme Lord Sri BhuVaraha removes this anxiety by promising that HE assumes the burden of remembering the JeevAthma.
Adiyen (ElayaAlwar) Srinivasa (DhoddayacharyAr) Dasan.