Can $T$ defined below be shown to be a completely positive map? $$T(A)=E[(I-XX')A(I-XX')]$$
$T$ maps between covariance matrices, X is a (column)-vector sampled from some $d$-dimensional distribution (ie, multivariate normal), $E$ is expectation over this distribution.
I've verified empirically that the Choi matrix is positive for a few empirical distributions of $X$ in 3 dimensions, does this hold more generally?
Every conditional expectation is CP (completely positive) by [1, IV.3.4], and every map of the form $A\mapsto B^*AB$ is clearly CP, so your map is a composition of CP maps, hence CP.
[1] Takesaki, M., Theory of operator algebras I., Encyclopaedia of Mathematical Sciences 124. Operator Algebras and Non-Commutative Geometry 5. Berlin: Springer (ISBN 3-540-42248-X/hbk). xix, 415 p. (2002). ZBL0990.46034.