The higher-dimensional singular value pressure inequality

Let $d\geq 1$. For a measure $\mu$ on $M_d(\mathbb{R})$, let $\mu_{n}$ denote its $n$-fold convolution and let $\varphi^s$ and $\mathbf{P}(\mu,s)$ denote the singular value function and matrix pressure, respectively. The condition $\int \varphi^s(A)\,d\mu(A)<\infty$ is assumed.

Higher-dimensional singular value pressure conjecture. For each $d\geq 1$, there exist an integer $n(d)$ and a continuous function $\hat{K}_d\colon(0,+\infty)\to(0,+\infty)$ such that, for every such measure $\mu$ and every $s>0$,

$$\int \varphi^s(A)\,d\mu_{n(d)}(A)\leq \hat{K}_d(s)e^{\mathbf{P}(\mu,s)}\left(\int \varphi^s(A)\,d\mu(A)\right)^{n(d)-1}.$$

This conjecture extends the established two-dimensional inequality to arbitrary matrix dimensions. The source states that its methods do not appear sufficient to prove the analogous inequality when $d>2$, so the proposed extension remains open.