  Let a $2\pi$-periodic function $f\in\Bbb C$ changes its monotonicity at a
finitely even number of points $y_i$ of the period. The degree of approximation
of this $f$ by trigonometric polynomials which are comonotone with it, i.e.
that change their monotonicity exactly at the points $y_i$ where $f$ does, is
restricted by $\omega_2(f,\pi/n)$ (with a constant depending on the location of
these $y_i$). Recently, we proved that relaxing the comonotonicity requirement
in intervals of length proportional to $\pi/n$ about the points $y_i$ (so
called nearly comonotone approximation) allows the polynomials to achieve the
approximation rate of $\omega_3$. By constructing a counterexample, we show
here that even with the relaxation of the requirement of comonotonicity for the
polynomials on sets with measures approaching $0$ (no matter how slowly or how
fast) $\omega_4$ is not reachable.
