Combinatorial invariance conjecture for ${\tilde{A}}_2$

The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x^{\mathrm{\prime }},y^{\mathrm{\prime }}]$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x^{\mathrm{\prime }},y^{\mathrm{\prime }}}(q)$. We prove this conjecture for the affine Weyl group of type ${\tilde{A}}_2$. This is the first infinite group with non-trivial Kazhdan-Lusztig polynomials where the conjecture is proved.