A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case $p\>2$ (but with no restrictions on the characteristic of $K$). In this paper...