Adolphson and Sperber characterized the unique unit root of the L-function associated with toric exponential sums in terms of the \(\mathcal {A}\) -hypergeometric functions. For the unit root L-function associated with a family of toric exponential sums, Haessig and Sperber conjectured that its unit root behaves similarly to the classical case studied by Adolphson and Sperber. Under the assumption of a lower deformation hypothesis, Haessig and Sperber proved this conjecture. In this paper, we demonstrate that Haessig and Sperber’s conjecture holds in general.
