Rayleigh quotient of a vector permutation

Question

The Rayleigh quotient of a Hermitian matrix $M$ and a complex vector $\mathbf{x}$ is defined as $$R(M,\mathbf{x})=\frac{\mathbf{x}^HM\mathbf{x}}{\mathbf{x}^H\mathbf{x}},$$ which if $\lambda_\text{max}$ and $\lambda_\text{min}$ denote the maximum and the minimum of the eigenvalues of $M$, respectively, then we have $$\lambda_\text{min}\leq R(M,\mathbf{x})\leq\lambda_\text{max}.$$ My question is either of the following equivalent statements. Let $P$ denotes a permutation matrix.

1. Given $R(M,\mathbf{x})=v$, does it imply that $R(M,P\mathbf{x})$ belongs to a proper subset of $[\lambda_\text{min},\lambda_\text{max}]$?
2. Given $R(M,\mathbf{x})=v$, can $R(M,P\mathbf{x})$ assume any value in $[\lambda_\text{min},\lambda_\text{max}]$?
3. Is there any non-empty $S_v\subset[\lambda_\text{min},\lambda_\text{max}]$ such that $R(M,P\mathbf{x})\notin S_v$?
4. Is it true that $\left\{\Big(R(M,\mathbf{x}),R(M,P\mathbf{x})\Big):\mathbf{x}\in\mathbf{C}^n\right\}\neq [\lambda_\text{min},\lambda_\text{max}]\times[\lambda_\text{min},\lambda_\text{max}]$?

Example: Assume $M$ is diagonal. Then $R(M,\mathbf{x})\in\text{diag}(M)$ implies that for any permutation matrix $P$ we have $R(M,P\mathbf{x})\in\text{diag}(M)$; thus $S_v=[\lambda_\text{min},\lambda_\text{max}]\backslash\text{diag}(M)$.