Robin spectral rigidity of strictly convex domains with a reflectional symmetry

This is a note on a recent paper of De Simoi-Kaloshin-Wei \cite{DKW}. We show that using their results combined with wave trace invariants of Guillemin-Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of \cite{DKW} to the case of Robin boundary conditions. We will consider the same generic subset as in \cite{DKW} of smooth strictly convex $\mathbb Z_2$-symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary $\mathbb Z_2$-symmetric smooth Robin functions on the boundary and of course allow deformations of Robin functions as well.