Projection Constants and Banach-Mazur Distance in Codimension Two

We prove that every subspace $V\subset\ell_\infty^N$ of codimension one or two with absolute projection constant $λ(V)<4/3$ satisfies $$ d(V,\ell_\infty^{\dim V}) \leq\frac{λ(V)}{4-3λ(V)}. $$ We conjecture that the codimension assumption can be removed and that the same estimate holds for every su