6. Let $a, b, c$ be sides of the triangle. Let $A_{1}$ be the intersection of line $A I$ with $B C$. By the known fact, $B A_{1}: A_{1} C=c: b$ and $A I: I A_{1}=A B: B A_{1}$, hence $B A_{1}=\frac{a c}{b+c}$ and $\frac{A I}{I A_{1}}=\frac{A B}{B A_{1}}=\frac{b+c}{a}$. Consequently $\frac{A I}{l_{A}}=\frac{b+c}{a+b+c}$. Put $a=n+p, b=p+m, c=m+n$ : it is obvious that $m, n, p$ are positive. Our inequality becomes
$$22 T^{3}$$
Remark. The inequalities cannot be improved. In fact, $\frac{A I \cdot B I \cdot C I}{l_{A} l_{B} l_{C}}$ is equal to $8 / 27$ for $a=b=c$, while it can be arbitrarily close to $1 / 4$ if $a=b$ and $c$ is sufficiently small.