Prof. Bang-Yen Chen, I do not know, and it seems quite tough. However, I would like to ask about the right-hand side. In particular, is the inequality sharp? Can equality be achieved? Also, as a remark, the right-hand side does not depend on $m$, which makes the conjecture even more interesting. Perhaps the right-hand side is related to some characteristic numbers of $HP^n$?

Prof. Bang-Yen Chen, are similar lower bounds on the total mean curvature (raised to some power) known to hold for some other compact manifolds? Do they hold for arbitrary compact Wolf spaces? For instance, in dimension $4$, does it hold for $CP^2$? I am not familiar at all with such problems, so I am just asking naively.

Dear Professor Malkoun:

There are similar conjectures and results for CP^n and RP^n (see my questions  also posed in RG).

I do not know any related conjecture on Wolf space.

Sincerely,

B.-Y. Chen

Professor Malkoun:

I shall mention that the most general result on total mean curvature is the following:

For every n-dimensional compact submanifold M immersed in a Euclidean m-space, the total mean curvature satisfies

∫M |H|n dV  ≥ cn,

where cn is the volume of n-sphere. 

You can find other results on total mean curvature in my books  "Total mean curvature and submanifolds of finite type", 1984 and 2015 editions.

Sincerely,

B.-Y. Chen