Monotonicity conjecture for average singular values of Gaussian matrices

Let $\alpha_{\mathbb{R}}(d)$ and $\alpha_{\mathbb{C}}(d)$ be the average singular value of a $d\times d$ matrix with random i.i.d. real-valued and complex-valued entries, respectively, distributed as $\mathcal{N}\left(0,\frac1d\right)$. Then, for all $d\geq 1$,

$$\alpha_{\mathbb{C}}(d+1) \leq \alpha_{\mathbb{C}}(d) \quad\text{and}\quad \alpha_{\mathbb{R}}(d+1) \geq \alpha_{\mathbb{R}}(d).$$

This conjecture concerns the contrasting dimension dependence of the normalized average singular value in the complex and real Gaussian ensembles; the paper motivates it through numerical computations and asymptotic bounds, but does not establish the claimed monotonicity for all dimensions.