Let $K$ be a positive integer and $C$ be any $K \times K$ non-singular matrix. For positive real numbers $q_1, \dots, q_K$, define $$\Sigma(q_1, \dots, q_K) = CC' + diag(\frac{1}{q_1}, \dots, \frac{1}{q_K})$$ and $$V(q_1, \dots, q_K) = I_k - C'\Sigma^{-1}C.$$

**Question:** Are each of the diagonal entries of $V$ log-convex in the arguments $q_1, \dots, q_K$?

The statistical interpretation is the following: suppose a decision maker is learning about $K$ unknowns $\theta_1, \dots, \theta_K$, which are i.i.d. normally distributed according to the prior. She has access to $K$ signals $X_1, \dots, X_K$, which are linear combinations of the unknowns (with coefficients $C$) plus i.i.d. noise. Then $V(q_1, \dots, q_K)$ represents her conditional covariance matrix given $q_k$ observations of signal $X_k$.

It's not too difficult to show that $V_{ii}$ is convex. By brute-force computation I found that it is in fact log-convex when $K = 2$. Thus I'm wondering whether that might be true in general (and why).