The answer to this question is yes if the metric on the complex projective space is of standard one and the immersion is isometric (cf. [B.-Y. Chen, total curvature of immersed manifolds. IV. Spectrum and total mean curvature. Bull. Inst. Math. Acad. Sinica 7 (1979), no. 3, 301–311]).
My conjecture to this question is yes, i.e., I conjectured that the inequality holds for every immersion in a Euclidean space with arbitrary codimension. See page 132 of my 2015 book [Total mean curvature and submanifolds of finite type, 2nd edition] published by World Scientific.