Is the Gromov conjecture still open? Today I read about Gromov's definition of minimal volume for smooth manifolds.
$$\min {\rm Vol}(M):=\inf_{|K_g|\leq1}\{{\rm Vol}(M,g)\}.$$
Gromov's conjecture states that for every closed simply connected odd-dimensional manifold $\min {\rm Vol}(M)=0$. Is the Gromov conjecture still open? Can anybody give an example for this conjecture?
Update: My teacher told to me that this conjecture can be solved by the Ricci flow method but I don't know how to use it. Can anybody give me an explanation to this? and how it works?
Thanks.
 A: This response is unfortunately not about simply-connected manifolds, I overlooked that assumption by the author of the question. So, this is not an answer. See my comment below.
Gromov writes in this paper (see Section 0.4) an example of an odd dimensional (not simply-connected) manifold with non-vanishing minimal volume, by constructing a lower bound involving the so-called simplicial volume (and a corresponding metric $g$).
A: According to wikipedia, the conjecture is still open. I do not see at all how RF can help here. (Apart from dimension 3 when the statement is of course a corollary of Perelman's geometrization theorem.) There is no positivity assumption on curvature(s) of $M$ in this conjecture and the Ricci Flow  in higher dimensions is mostly a mystery without some positivity assumptions. From what you wrote, it sounds like your teacher has no idea how to approach this problem via Ricci Flow either; I suggest you work on something else, more doable. 
One more thing: The conjecture is only in odd-dimensional case. For even dimensional manifolds Gromov noted in his paper that there is a bound $MinVol(M^n)\ge c_n|\chi(M)|$, $c_n>0$. Hence, all even-dimensional spheres have positive minimal volume.
Edited: (4 September 2017) 
An Example of vanishing minimal volume which is due to Gromov is three-sphere. (see Gromove paper,section 0.4)
