Lin's multivariable matrix Young conjecture

Let $M_n$ denote the set of $n$ by $n$ complex matrices. For $X\in M_n$, write

$$|X|=(X^*X)^{1/2}.$$

For Hermitian matrices $X$ and $Y$, write $X\preceq Y$ when $Y-X$ is positive semidefinite. Let $A_1,\ldots,A_m\in M_n$ be positive semidefinite, and let $p_1,\ldots,p_m>0$ satisfy

$$\frac1{p_1}+\cdots+\frac1{p_m}=1.$$

The eigenvalues of a Hermitian matrix $H$ are ordered as $\lambda_1(H)\geq\lambda_2(H)\geq\cdots\geq\lambda_n(H)$.

Lin's conjecture. There exists a unitary matrix $U\in M_n$ such that

$$U|A_1\cdots A_m|U^*\preceq \frac1{p_1}A_1^{p_1}+\cdots+\frac1{p_m}A_m^{p_m}.$$

Equivalently, for $j=1,\ldots,n$,

$$\lambda_j(|A_1\cdots A_m|) \leq \lambda_j\left( \frac1{p_1}A_1^{p_1}+\cdots+\frac1{p_m}A_m^{p_m} \right).$$

This is a multivariable extension of matrix Young inequalities in which the order of the factors matters. The conjecture was proposed by Lin, but the paper shows that it fails for every $m\geq3$: for $m\geq4$ there are uniform $2\times2$ counterexamples, while $m=3$ is disproved by a $3\times3$ positive semidefinite example.