  Exploration of the intractable posterior distributions associated with
Bayesian versions of the general linear mixed model is often performed using
Markov chain Monte Carlo. In particular, if a conditionally conjugate prior is
used, then there is a simple two-block Gibbs sampler available. Rom\'{a}n and
Hobert [Linear Algebra Appl. 473 (2015) 54-77] showed that, when the priors are
proper and the $X$ matrix has full column rank, the Markov chains underlying
these Gibbs samplers are nearly always geometrically ergodic. In this paper,
Rom\'{a}n and Hobert's (2015) result is extended by allowing improper priors on
the variance components, and, more importantly, by removing all assumptions on
the $X$ matrix. So, not only is $X$ allowed to be (column) rank deficient,
which provides additional flexibility in parameterizing the fixed effects, it
is also allowed to have more columns than rows, which is necessary in the
increasingly important situation where $p>N$. The full rank assumption on $X$
is at the heart of Rom\'{a}n and Hobert's (2015) proof. Consequently, the
extension to unrestricted $X$ requires a substantially different analysis.
