Monotonicity conjecture for separability of random induced states

Let $d\geqslant 2$ and let $\pi_{d,s}$ denote the probability that a random state on $\mathbf{C}^d\otimes\mathbf{C}^d$ with distribution $\mu_{d^2,s}$ is separable. Monotonicity conjecture. For any $d\geqslant 2$, the function $s\mapsto\pi_{d,s}$ is non-increasing. This conjecture formalizes the intuition that increasing the environment dimension makes entanglement rarer; it would improve the paper's probability estimates and imply a bound of the form $\pi_{d,s}\leqslant 2\exp(-c(\varepsilon)s_0(d))$.