  Let $A$ be a uniform algebra, $\theta:A\to M_n(\mathbb{C})$ be a continuous
homomorphism and $\alpha:A\to A$ be an antilinear contraction such that \[
\|\theta(f)+\theta(\alpha(f))^*\|\le 2\|f\| \quad(f\in A). \] We show that
$\|\theta\|\le 1+\sqrt{2}$, and that $1+\sqrt2$ is sharp. We conjecture that,
if further $\alpha(1)=1$, then we may conclude that $\|\theta\|\le2$. This
would yield a positive solution to the Crouzeix conjecture on numerical ranges.
In support of our conjecture, we prove that it is true in two special cases. We
also discuss a completely bounded version of our conjecture that brings into
play ideas from dilation theory.
