Product conjecture for Patnaik–Pearson dimension of random matrix products

Let $c{\lambda}$ and $c{\phi}$ be regularly varying random variables, with tail exponents $calpha$ and $cbeta$, respectively. Let $A$ and $B$ be random matrices in general position, of dimensions $N ctimes d$ and $d ctimes m$, respectively, with $N$, $d$, and $m$ all sufficiently large. Suppose that their singular-value decompositions are

$$A=U_A\operatorname{diag}(\boldsymbol{\lambda})V_A^t,\qquad B=U_B\operatorname{diag}(\boldsymbol{\phi})V_B^t,$$

where the singular values are iid with $clambda_i\simclambda$ and $cphi_i\simcphi$. Considering $coperatorname{PP}(A)$, $coperatorname{PP}(B)$, and $coperatorname{PP}(AB)$ as random variables, Product conjecture.

$$\frac{1}{d}\mathbb{E}(\operatorname{PP}(A))\,\frac{1}{m}\mathbb{E}(\operatorname{PP}(B))\leq\frac{1}{m}\mathbb{E}(\operatorname{PP}(AB))\leq\min\left\{\frac{1}{d}\mathbb{E}(\operatorname{PP}(A)),\frac{1}{m}\mathbb{E}(\operatorname{PP}(B))\right\}.$$

The conjecture predicts that multiplying random matrices combines their normalized intrinsic dimensions multiplicatively as a lower bound, while the product cannot exceed the smaller normalized factor. The source motivates it with numerical experiments and a heuristic based on regularly varying singular values; a rigorous derivation is suggested via free-probability transforms, but no resolution is given.