= Solution
Because the condition is local and covariant, choose <normal coordinates> at the point and a Lorentz frame in which the timelike observer is $X^\mu=\delta^\mu_0$. Rescaling a timelike vector by a positive factor does not affect the sign, so it suffices to use this unit vector. With signature $(-,+,\ldots,+)$,
$$
\nabla_\rho\phi\nabla^\rho\phi
=-(\partial_0\phi)^2+\sum_{i=1}^{D-1}(\partial_i\phi)^2.
$$
The measured energy density is therefore
$$
\begin{aligned}
2T_{\mu\nu}X^\mu X^\nu
=2T_{00}
&=(\partial_0\phi)^2
-A\left[-(\partial_0\phi)^2+\sum_i(\partial_i\phi)^2\right]
-B\phi^2\\
&=(1+A)(\partial_0\phi)^2
-A\sum_i(\partial_i\phi)^2-B\phi^2.
\end{aligned}
$$
The value of $\phi$, its time derivative, and its spatial first derivatives can be varied independently at a point. Nonnegativity for all such data is therefore equivalent to nonnegativity of all three coefficients:
$$
1+A\geq0,
\qquad
-A\geq0,
\qquad
-B\geq0.
$$
Thus the most general constraints are
$$
\boxed{-1\leq A\leq0,
\qquad B\leq0}.
$$
These conditions are also sufficient in every timelike frame, because any timelike vector can be brought to the chosen rest frame. This is the <weak energy condition for a quadratic scalar stress-energy ansatz>.
The conserved values from part (a),
$$
A=-\frac12,
\qquad
B=-\frac{m^2}{2},
$$
satisfy these inequalities for real $m$, as required.
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