  We consider the analogue of Rayleigh's conjecture for the clamped plate in
Euclidean space weighted by a log-convex density. We show that the lowest
eigenvalue of the bi-Laplace operator with drift in a given domain is bounded
below by a constant $C(V,n)$ times the lowest eigenvalue of a centered ball of
the same volume; the constant depends on the volume $V$ of the domain and the
dimension $n$ of the ambient space. Our result is driven by a comparison
theorem in the spirit of Talenti, and the constant $C(V,n)$ is defined in terms
of a minimization problem following the work of Ashbaugh and Benguria. When the
density is an "anti-Gaussian," we estimate $C(V,n)$ using a delicate analysis
that involves confluent hypergeometric functions, and we illustrate numerically
that $C(V,n)$ is close to $1$ for low dimensions.
