Loewner's conjecture is the local assertion that an isolated umbilic point of a smooth surface in 
has principal-direction index at most 1. The index measures
the turning of an unoriented principal direction
line field around the point (Gutiérrez and Sánchez-Bringas 1997). This
bound would imply Carathéodory's conjecture
for a closed strictly convex surface.
Ando and Umehara (2026) reported smooth non-real-analytic local constructions with indices 
for arbitrarily large positive integers 
, as well as a global one-umbilic construction. Independent
verification of the full constructions has not been established, so the claimed counterexamples
remain qualified. Ando and Umehara (2026) report AI assistance with searching for
candidate functions in the mathematical construction.
