What is a tau-function?

We conjecture that every local holomorphic (with respect to both variables) solution of the Korteweg–de Vries equation is the second logarithmic derivative (for every fixed value of the time variable and up to a coefficient) of an entire function of order at most 3 in the space variable. Besides motivation, particular cases and corollaries of this conjecture, we also consider an attempt to define this entire function by means of the coefficients in (an appropriate generalization of) the Szegő–Widom asymptotic formula for the determinants of truncated Toeplitz matrices.