Orthogonal similarity of block real symmetric matrix

Question

Can a block real symmetric matrix whose sum of diagonal blocks is an identity matrix $$\left[ \begin{array}{cccc} H^{T}_{1}H_{1} & H^{T}_{2}H_{2} & \cdots & H^{T}_{1}H_{n+1} \\ H^{T}_{1}H_{2} & H^{T}_{2}H_{2} & \cdots & H^{T}_{2}H_{n+1} \\ \vdots & \vdots & & \vdots \\ H^{T}_{1}H_{n+1} & H^{T}_{2}H_{n+1} & \cdots & H^{T}_{n+1}H_{n+1} \\ \end{array} \right]~\left(H^{T}_{1}H_{1}+\cdots+H^{T}_{n+1}H_{n+1}=I_{N}\right)$$ be orthogonally similar to a block diagonal matrix whose sum of diagonal blocks is also an identity matrix $$\left[ \begin{array}{cccc} \tilde{H}^{T}_{1}\tilde{H}_{1} & & & \\ & \tilde{H}^{T}_{2}\tilde{H}_{2} & & \\ & & \ddots & \\ & & & \tilde{H}^{T}_{n+1}\tilde{H}_{n+1} \\ \end{array} \right]~\left(\tilde{H}^{T}_{1}\tilde{H}_{1}+\cdots+\tilde{H}^{T}_{n+1}\tilde{H}_{n+1}=I_{N}\right)$$