Generalizing the interlacing property of eigenvalues for matrix pencil

Question

Let $A_1, \dots, A_k, B_1, \dots, B_k\in\mathbb{R}^{n\times n}$ be symmetric positive definite matrices, and suppose that they all commute with each other. Let $U\in\mathbb{R}^{n\times r}$ be a matrix with orthonormal columns, i.e. such that $U^\top U = I_r$, and let us define $\widetilde{A}_i = U^\top A_i U$ and $\widetilde{B}_i = U^\top B_i U$ for all $i = 1,\dots, k$. From the interlacing properties of eigenvalues for definite matrix pencils we know that it holds $\rho(\widetilde{A}_i^{-1}\widetilde{B}_i) \leq \rho(A_i^{-1}B_i)$ for all $i=1,\dots, k$, where $\rho(\cdot)$ denotes the spectral radius. Does it also hold the following? \begin{equation} \rho(\widetilde{A}_1^{-1}\widetilde{B}_1 \widetilde{A}_2^{-1}\widetilde{B}_2 \cdots\widetilde{A}_k^{-1}\widetilde{B}_k) \leq \rho(A_1^{-1}B_1A_2^{-1}B_2\cdots A_k^{-1}B_k). \end{equation} Observe that in general the matrices $\{\widetilde{A}_i, \widetilde{B}_i\}_{i=1}^k$ do not commute.