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.

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)

Let me convert my comment to an answer:

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.

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.