Fix $0<h_1<h_2<h_3<1$ reals. All matrices below are $3\times3$ real.

Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric positive definite matrix $M$ (point-wise). Assume that the eigenvalues of $M(n)$ converge to that of $M$.

QUESTION.Is it true that the 2nd eigenvalues of the product $$A(n):=\begin{pmatrix} n^{h_3-h_1}&0&0 \\ 0&1&0 \\ 0&0& n^{h_1-h_2} \end{pmatrix}\,M(n)\, \begin{pmatrix} n^{h_3-h_1}&0&0 \\ 0&1&0 \\ 0&0& n^{h_1-h_2} \end{pmatrix}$$ converge (to a finite number)?

The 1st eigenvalues of $A(n)$ diverge to $\infty$ while the 3rd eigenvalues of $A(n)$ converge to $0$, as $n\rightarrow\infty$. It seems that the 2nd eigenvalues can be shown to be bounded.