Constraints for Type 3 Linear Phase Filter
There is no such restriction for a type III linear phase filter. Neither is there such a restriction for any other type of linear phase filter. Type III linear phase filters have an odd filter length $N$ and odd symmetry. I.e., as long as the filter's impulse response $h[n]$ satisfies

$$h\left[\frac{N-1}{2}+n\right]=-h\left[\frac{N-1}{2}-n\right]$$

you have a proper type III linear phase filter.

If you look at the impulse responses of type III differentiators or Hilbert transformers then you'll notice that in both cases the filter taps are decreasing in magnitude as you move away from the center tap. So these common applications of type III filters are counterexamples to your assumption.