That the inequality you state is not true is discussed here, in fact: mathoverflow.net/questions/89436/…

...(cont) Berman-Berndtsson's paper "The projective space has maximal volume among all toric Kaehler-Einstein Manifolds" shows that for $X$ toric and Kaehler-Einstein, one have $vol(X)\leq (n+1)^n$. I don't know if $vol(X) = (n+1)^n$ implies $X = \mathbb{P}^n$, however. Sorry if I've misunderstood! I'll add this as an answer provided I haven't misunderstood the question.

I'm not an expert, so I'm hesitant to add this as an answer. Doesn't the bound $vol(X) \leq (n+1)^n.\frac{n+1}{I(n)}$ only hold for Kaehler-Einstein $X$? In Debarre's Higher dimensional algebraic geometry (pg 139), it is shown that there is no polynomial upper bound on $vol(X)$, even for toric $X$... (cont)

...(cont) I suspect when $X_r^n$ is Fano, one can similarly explicitly describe the nef cone (following the cone theorem), however I'm not sure how helpful that is. For $r \geq 9$ and $n=2$, the nef cone isn't known explicitly (this is related to Nagata's conjecture). The point (I think) is $X$ is no longer Fano, and there are infinitely many exceptional curves, as $9$ points determine a cubic.

Probably you know the following. For $n=2$ and $r \leq 8$, one knows explicitly the exceptional curves (Manin, Cubic Forms, Theorem 26.2). Thus one can can check explicitly if a divisor is nef, by intersecting with the exceptional curves. Then for surfaces, the nef cone is dual to the effective cone, so one can explicitly check if a divisor is effective (cont)...

I haven't read the paper, so this is just a comment, but a paper of Fujino on the arXiv today claims to show that the canonical ring of a compact Kaehler manifold is finitely generated: arXiv:1309.3015 (Corollary 4.2).