The value at the mode in multivariate t distributions: a curiosity or not?

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It is a well-known fact that multivariate Student t distributions converge to multivariate Gaussian distributions as the number of degrees of freedom ν tends to infinity, irrespective of the dimension k≥1. In particular, the Student's value at the mode (that is, the normalizing constant obtained by evaluating the density at the center) cν,ₖ=Γ(ν+k/2)(πν)ᵏ/² Γ(ν2) converges towards the Gaussian value at the mode cₖ=1(2π)ᵏ/². In this note, we prove a curious fact: cν,ₖ tends monotonically to cₖ for each k, but the monotonicity changes from increasing in dimension k=1 to decreasing in dimensions k≥3 whilst being constant in dimension k=2. A brief discussion raises the question whether this a priori curious finding is a curiosity, in fine.