Entropy inequality of joint von Neumann entropy given some marginal inequalities
Why should this be true?

Let the dimensions of $A, B, C, D$ all be the same. 

Let $\omega_{AB}$ and $\omega_{CD}$ both be pure states, with 
$$S(\omega_A) = S(\omega_B) = S(\omega_C) = S(\omega_D).$$

Now, let $U$ perform a cyclic permutation on $A, B, C, D$. 

We have $S(\tau_{AB}) = 2 S(\tau_B)$, and  $S(\omega_{AB}) = 0$, while
$S(\tau_B) = S(\omega_B)$. 

Now all you have to do is show that you can perturb this construction slightly to get the desired inequalities in your hypothesis.