  The weighted $k$-server problem is a natural generalization of the $k$-server
problem in which the cost incurred in moving a server is the distance traveled
times the weight of the server. Even after almost three decades since the
seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem
remains poorly understood, even on the simplest class of metric spaces -- the
uniform metric spaces. In particular, in the case of randomized algorithms
against the oblivious adversary, neither a better upper bound that the doubly
exponential deterministic upper bound, nor a better lower bound than the
logarithmic lower bound of unweighted $k$-server, is known. In this article, we
make significant progress towards understanding the randomized competitive
ratio of weighted $k$-server on uniform metrics. We cut down the triply
exponential gap between the upper and lower bound to a singly exponential gap
by proving that the competitive ratio is at least exponential in $k$,
substantially improving on the previously known lower bound of about $\ln k$.
