  We analyze the Landau-Wilson field theory with $\text{U}(n)\times\text{U}(m)$
symmetry which describes the finite-temperature phase transition in QCD in the
limit of vanishing quark masses with $n=m=N_f$ flavors and unbroken anomaly at
the critical temperature. The six-loop expansions of the renormalization group
functions are calculated within the Minimal Subtraction scheme in $4 -
\varepsilon$ dimensions. The $\varepsilon$ series for the upper marginal
dimensionality $n^{+}(m,4-\varepsilon)$ -- the key quantity of the theory --
are obtained and resummed by means of different approaches. The numbers found
are compared with their counterparts obtained earlier within lower perturbative
orders and the pseudo-$\varepsilon$ analysis of massive six-loop
three-dimensional expansions. In particular, using an increase in the accuracy
of numerical results for $n^{+}(m,3)$ by one order of magnitude, we strengthen
the conclusions obtained within previous order in perturbation theory about
fairness of the inequality $n^{+}(m,3)>m$. This, in turn, indicates the absence
of a stable three-dimensional fixed point for $n=m$, and as a consequence a
first-order kind of finite-temperature phase transition in light QCD.
