Can we decompose a square matrix with singular values smaller than 1 into a product of symmetric matrices with singular values smaller than 1?

Question

It is known that one can decompose a matrix $A$ into a product of two symmetric matrices $S_1,S_2$ (see Bosch, 1987 https://www.jstor.org/stable/2031239 for instance, using Jordan decomposition)

But given that the greatest singular value of $A$ is smaller than $1$, can we find such $S_1$ and $S_2$ such that the singular values of $S_1$ and $S_2$ are smaller than $1$?