  The Rayleigh Conjecture for the bilaplacian consists in showing that the
clamped plate with least principal eigenvalue is the ball. The conjecture has
been shown to hold in 1995 by Nadirashvili in dimension $2$ and by Ashbaugh and
Benguria in dimension $3$. Since then, the conjecture remains open in dimension
$d\geq 4$. In this paper, we contribute to answer this question, and show that
the conjecture is true in any dimension as long as some special condition holds
on the principal eigenfunction of an optimal shape. This condition regards the
mean value of the eigenfunction, asking it to be in some sense minimal. This
main result is based on an order reduction principle allowing to convert the
initial fourth order linear problem into a second order affine problem, for
which the classical machinery of shape optimization and elliptic theory is
available. The order reduction principle turns out to be a general tool. In
particular, it is used to derive another sufficient condition for the
conjecture to hold, which is a second main result. This condition requires the
Laplacian of the optimal eigenfunction to have constant normal derivative on
the boundary. Besides our main two results, we detail shape derivation tools
allowing to prove simplicity for the principal eigenvalue of an optimal shape
and to derive optimality conditions. Eventually, because our first result
involves the principal eigenfunction of a ball, we are led to compute it
explicitly.
