Relations of the Trace Energy Condition to other energy conditions From 1935 to the 60's, there was in common use the trace energy condition (or TEC), which stated that for any stress energy tensor, $\text{Tr}(T) = T_{\mu\nu} g^{\mu\nu} \leq 0$ was a reasonable condition. This is due to an argument of von Neumann in Chandrasekhar's paper "Stellar configurations with degenerate cores". I don't know the actual argument since it was apparently never published (I think it's related to free gas of particles).
Due to its fall from use, I don't think I've ever seen any proof relating to its relations with other energy conditions. If we use the stress energy tensor of a perfect fluid, this corresponds to the condition $\rho + 3p \geq 0$, which suggests that it should at least imply the null energy condition. 
I'm not quite sure how to perform this proof. Here's a few attempts : 
$1)$ By the trace of the EFE 
The trace of the EFE is famously 
$$T = (1 - \frac{n}{2})R$$
which means that for every $n > 2$ ($T = 0$ for any spacetime in $1+1$ dimensions, so we can safely ignore it), 
$$T \leq 0 \leftrightarrow R \geq 0$$
Unfortunartely, this doesn't really offer any insight in any relations with other energy conditions.
$2)$ Decomposition of the metric tensor
Any Lorentzian metric tensor can famously be decomposed into a Riemannian metric tensor $h$ and a (nowhere vanishing) timelike vector field $\xi$
$$g_{\mu\nu} = h_{\mu\nu} - 2\frac{\xi_\mu \xi_\nu}{h(\xi,\xi)}$$
The TEC then becomes 
$$T^{\mu\nu} h_{\mu\nu} - 2 T^{\mu\nu}\frac{\xi_\mu \xi_\nu}{h(\xi,\xi)}$$
Assuming that $\xi_\mu$ are still timelike (Since they are actually $\xi^\mu h_{\mu\nu}$ I'm not 100% sure), we should have 
$$T^{\mu\nu} h_{\mu\nu} \leq 2 T^{\mu\nu}\frac{\xi_\mu \xi_\nu}{h(\xi,\xi)}$$
This depends on the Riemannian trace of $T$, which does not seem to help me all that much.
$3)$ Segre classification of the stress energy tensor
In 4 dimensions, there are only 4 types of stress energy tensor possibles . These are : 
Type I:
$$ T^{\mu\nu} = \begin{pmatrix}
\rho & 0 & 0 & 0\\
0 & p_1 & 0 & 0\\
0 & 0 & p_2 & 0\\
0 & 0 & 0 & p_3
 \end{pmatrix}$$
Type II : 
$$ T^{\mu\nu} = \begin{pmatrix}
\mu + f & f & 0 & 0\\
f & -\mu + f & 0 & 0\\
0 & 0 & p_2 & 0\\
0 & 0 & 0 & p_3
 \end{pmatrix}$$
Type III : 
$$ T_{\mu\nu} = \begin{pmatrix}
\rho & 0 & f & 0\\
0 & -\rho & f & 0\\
f & f & -\rho & 0\\
0 & 0 & 0 & p
 \end{pmatrix}$$
and Type IV : 
$$ T^{\mu\nu} = \begin{pmatrix}
\rho & f & 0 & 0\\
f & -\rho & 0 & 0\\
0 & 0 & p_1 & 0\\
0 & 0 & 0 & p_2
 \end{pmatrix}$$
They can all be combined together under the form
$$ T^{\mu\nu} = \begin{pmatrix}
\rho & f_1 & f_2 & f_3\\
f_1 & p_1 & 0 & 0\\
f_2 & 0 & p_2 & 0\\
f_3 & 0 & 0 & p_3
 \end{pmatrix}$$
The TEC corresponds then to $-\rho + p_1 + p_2 + p_3 \leq 0$, or 
$$ p_1 + p_2 + p_3 \leq \rho$$
which might imply that the SEC implies the TEC, but that's a bit tough to show (the general expression of Segre types of the SEC is a bit messy), and of course that only holds in 4 dimensions. I think it might really disqualify the TEC from implying most other energy conditions, since that neither forbids negative energy density nor does it constrain individual components (for $p_1 \ll 0$ we can have the pressure in other direction arbitrarily negative).
What are the possible implications between the TEC and other energy conditions? That is, 
$1)$ If a spacetime obeys the TEC, does it obey any other energy condition?
$2)$ If a spacetime obeys another energy condition, does it obey the TEC?
 A: I am you from the future.
Here are a few counterexamples to various implications of the Trace Energy Condition, using type I stress energy tensors as an example : 
TEC $\to$ DEC : For $T = \text{diag}(\rho, 2\rho, 2\rho, 2\rho)$, the TEC is obeyed but not the DEC.
TEC $\to$ SEC : For $T = \text{diag}(\rho, -2\rho, 2\rho, 2\rho)$, the TEC is obeyed but not the SEC.
TEC $\to$ WEC : For $T = \text{diag}(-\rho, \rho, \rho, \rho)$, the TEC is obeyed but not the WEC.
TEC $\to$ NEC : For $T = \text{diag}(\rho, -2\rho, \rho, \rho)$, the TEC is obeyed but not the NEC.
The TEC does not hence imply any of the classical energy conditions (it can also be shown more simply by taking the type III stress energy tensor with $p = 2 \rho$, since it obeys the TEC but no classical energy condition obeys Type III, or just showing it for the NEC since all others imply it).
NEC $\to$ TEC : For the type II SET with $\mu = 3\rho$, the pressure $p_1, p_2 = -\rho$ and $f < \rho$, the NEC is obeyed but not the TEC. 
The same argument applies for the WEC and the DEC. I can't think of an counterexample for the implication SEC $\to$ TEC, though, it seems to hold for every type in 4 dimensions, so if a counterexample exists, it would have to be for $n > 4$. 
I'm not sure how well those counterexamples hold for averaged energy conditions but I think a few of them should hold at least.
A: Let's suppose a stress-energy tensor of type I (In the following description I will consider a signature -2.). For this case, the main (pointwise) energy conditions can be written as:

*

*Dominant energy condition (DEC)

\begin{equation}
\label{DEC}\rm{DEC1} \equiv  \rho  \geq 0, \ \rm{DEC2}_{i} \equiv \rho + p_{i} \geq 0 \ \ \text{and} \ \ \rm{DEC3}_{i} \equiv \rho - p_{i} \geq 0.
\end{equation}

*

*Weak energy condition (WEC)

\begin{equation}
\label{WEC}\rm{WEC1} \equiv  \rho  \geq 0 \ \ \text{and} \ \ \rm{WEC2}_{i} \equiv \rho + p_{i} \geq 0.
\end{equation}

*

*Strong energy condition (SEC)

\begin{equation}
\label{SEC}\rm{SEC1}_{i} \equiv  \rho + p_{i} \geq 0 \ \ \text{and} \ \ \rm{SEC2} \equiv \rho + \sum_{i}^{3} p_{i} \geq 0.
\end{equation}

*

*Null energy condition (SEC)

\begin{equation}
\label{NEC}\rm{NEC}_{i} \equiv  \rho + p_{i} \geq 0.
\end{equation}
For its turn, the Trace Energy Condition (TEC) as:
\begin{equation}
\label{TEC}\rm{TEC} \equiv  \rho - p_{1} - p_{2} - p_{3} \geq 0.
\end{equation}
As we can see, (1) if the spacetime obeys the TEC, it does not necessarily obey the other energy conditions; and (2) if the spacetime obeys any pointwise energy condition, it does not necessary obey the TEC. The effecfive form of the energy conditions are different. This conclusion holds at least for the stress-energy tensor of type I.
