  A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in
$\mathbb N$. A generalized numerical semigroup is a submonoid of
$\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of
numerical semigroups, Wilf's conjecture is a long standing open problem whose
study has led to new mathematics and new ways of thinking about monoids. A
natural extension of Wilf's conjecture, to the class of $\mathcal
C$-semigroups, was proposed by Garc\'ia-Garc\'ia, Mar\'in-Arag\'on, and
Vigneron-Tenorio. In this paper, we propose a different generalization of
Wilf's conjecture, to the setting of generalized numerical semigroups, and
prove the conjecture for several large families including the irreducible,
symmetric, and monomial case. We also discuss the relationship of our
conjecture to the extension proposed by Garc\'ia-Garc\'ia, Mar\'in-Arag\'on,
and Vigneron-Tenorio.
