  We investigate relativistic star solutions in Mass-Varying Massive Gravity
(MVMG) with a diagonal metric. Contrary to the intuition that there is no
fundamental difference between diagonal metric and non-diagonal metric
solutions regarding relativistic stars, we find that with a diagonal metric,
well-behaved relativistic star solutions may not exist except for trivial ones
in which the graviton mass is a constant, whereas non-trivial relativistic star
solutions had been found in MVMG with a non-diagonal metric. The reason is that
with a diagonal metric, the field equations constitute a system of
differential-algebraic equations of differential index-2 with two extra
constraints that have a significant influence on the system, rendering the
relativistic star solution with a non-trivial graviton mass configuration
impossible in most cases.
