Shub's conjecture on eigenvalue products and Grassmannian distortion

Let $A \in GL(d,\mathbb{R}) \setminus \mathbb{R}I$. For a Haar-random $U \in \mathrm{SO}(d)$, let $\lambda_1(UA),\ldots,\lambda_d(UA)$ denote the eigenvalues ordered by decreasing modulus. Let $g_k$ be a uniformly random $k$-dimensional Grassmannian in $\mathbb{C}^d$, represented by the span of the first $k$ columns of $U$, and write $U_k$ for the resulting $d \times k$ matrix and

$$\det A|g_k:=\det\bigl((AU_k)^*(AU_k)\bigr).$$

Shub's conjecture. There is a constant $c_{d,k}$ such that

$$\int_{\mathrm{SO}(d)}\prod_{i=1}^k|\lambda_i(UA)|\,d\nu(U)\geq c_{d,k}\int_{\mathrm{Grass}(d,k)}\det A|g_k\,d\mu(g_k),$$ $$\int_{\mathrm{SO}(d)}\sum_{i=1}^k\log|\lambda_i(UA)|\,d\nu(U)\geq c_{d,k}\int_{\mathrm{Grass}(d,k)}\log\det A|g_k\,d\mu(g_k),$$

and one can choose $c_{d,k}=1$ for all $d$ and $k$. The conjecture compares averaged products and logarithmic sums of the top eigenvalue moduli with averaged distortion on random Grassmannians. The full conjecture over $\mathrm{SU}(d)$ was proven by Dedieu and Shub, while the stated real orthogonal-group version is presented here as the question being addressed; the paper obtains asymptotic progress for the spectral radius.