  Suppose $g=g_0+g_1$ is a finite-dimensional restricted Lie superalgebra over
an algebraically closed field $k$ of characteristic $p>2$. In this article, we
propose a conjecture for maximal dimensions of irreducible modules over the
universal enveloping algebra $U(g)$ of $g$, as a super generalization of the
celebrated first Kac-Weisfeiler conjecture. It is demonstrated that the
conjecture holds for all basic classical Lie superalgebras and all completely
solvable restricted Lie superalgebras. In this process, we investigate
irreducible representations of solvable Lie superalgebras.
