Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ being solved? Let A be a positive semidefinite matrix of order $n$.
Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ is being solved? 
Where


*

*$\tilde{A}$ is the matrix of order $n!\times n!$whose $(\sigma, \tau)^{th}$ entry is given by $\prod _{i=1}^{n} a_{{\sigma (i)}\tau(i)}$

*$Per (A)=\sum _{\sigma\in S_n} \prod_{i=1}^{n} a_{i\sigma (i)}$


The conjecture has being proved for $n\leq 3$  in Bapat, R. B. ; Sunder, V. S. An extremal property of the permanent and
the determinant. Linear Algebra Appl. 76 (1986), 153–163. 
Is there any further development in this problem?

The permanent-on-top conjecture stating that per(A) is the largest eigenvalue of the Schur power matrix  is false! See a counterexample  for n=5 and rank = 2 here:
https://www.researchgate.net/publication/280495158_The_permanent_on_top_conjecture_is_false 
