By a result of the second author, the Connes embedding conjecture (CEC) is
false if and only if there exists a selfadjoint noncommutative polynomial
$p\left({t}_{1},{t}_{2}\right)$ in the universal
unital
${C}^{\ast}$algebra
$\mathcal{A}=\langle {t}_{1},{t}_{2}:{t}_{j}={t}_{j}^{\ast}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}0<{t}_{j}\le 1for1\le j\le 2\rangle $ and positive, invertible
contractions
${x}_{1},{x}_{2}$ in a finite
von Neumann algebra
$\mathcal{\mathcal{M}}$
with trace
$\tau $
such that
$\tau \left(p\left({x}_{1},{x}_{2}\right)\right)<0$ and
${Tr}_{k}\left(p\left({A}_{1},{A}_{2}\right)\right)\ge 0$ for every positive
integer
$k$ and all positive
definite contractions
${A}_{1},{A}_{2}$
in
${M}_{k}\left(\u2102\right)$. We prove
that if the real parts of all coefficients but the constant coefficient of a selfadjoint polynomial
$p\in \mathcal{A}$ have the same sign, then
such a
$p$ cannot disprove
CEC if the degree of
$p$
is less than
$6$,
and that if at least two of these signs differ, the degree of
$p$ is
$2$, the coefficient
of one of the
${t}_{i}^{2}$
is nonnegative and the real part of the coefficient of
${t}_{1}{t}_{2}$ is zero then
such a
$p$
disproves CEC only if either the coefficient of the corresponding linear term
${t}_{i}$ is nonnegative or both
of the coefficients of
${t}_{1}$
and
${t}_{2}$
are negative.
