(Edited solution slightly after OP edited the question) There are (still) some trivial answers.
Let $A$ be any TM which accepts the empty string and let $B$ be the TM, which instantly halts, except in the case of the empty string, in which case it goes into an infinite loop.
Now for each non-empty string $IF$ B does not halt, then $A $ will certainly halt. Since $B$ will halt, this is clearly true. For the non-empty string we know that $A$ halts and $B$ does not halt, thus the implication clearly hold for any input.
In the same spirit let $C$ be any TM which does not accept the empty string, and let $D$ be the TM which just loops and do not halt for any string, except for the empty string which it accepts imediately.