  We obtain new two-sided norm estimates for the family of Bergman-type
projections arising from the standard weights $(1-|z|^2)^{\alpha}$ where
$\alpha>-1$. As $\alpha\to -1$, the lower bound is sharp in the sense that it
asymptotically agrees with the norm of the Riesz projection. The upper bound is
estimated in terms of the maximal Bergman projection, whose exact operator norm
we calculate. The results provide evidence towards a conjecture that was posed
very recently by the first author.
