Bayesian: Does the choice of the prior on the covariance matrix influence the posterior mean in a gaussian model?

Question

Suppose we have a Multivariate Gaussian model, with some observed data $X$: $$X \; | \; \mu, \Sigma \sim MultiNormal(\mu, \Sigma)\\ \text{each }\mu_i \sim Normal(\mu_0, \sigma_0)$$ and some prior on $\Sigma$, e.g. Inverse Wishart.

My question is: Does the choice of the prior for $\Sigma$ influence the marginal posterior for $\mu_i | X$?

Intuitively I would say no, or at least not influencing the expected value $\mathbb{E} \; \mu_i | X$, however in [1] the posterior mean is dependent on $\Sigma$. Can you give me a intuitive explanation why?

[1] Multivariate normal posterior