Conjecture on Frobenius norms of projected bidiagonal blocks

Let ${\bf B}_{m+\ell}$ be partitioned as

$${\bf B}_{m+\ell}=\begin{bmatrix}{\bf B}_m & \begin{bmatrix}{\bf0}_{m\times\ell} \\ \alpha_{m+1}{\bf e}_1\tau\end{bmatrix} \\ {\bf0}_{\ell\times m} & \overline{{\bf B}}_\ell\end{bmatrix},$$

and define

$$\overline{\overline{{\bf B}}}_\ell=\begin{bmatrix}\alpha_{m+1}{\bf e}_1\tau \\ \overline{{\bf B}}_\ell\end{bmatrix}.$$

Here $\widetilde{{\bf B}}_\ell$ is the corresponding projected bidiagonal block produced by the recycled hybrid projection method. Frobenius-norm conjecture. The two blocks have approximately equal squared Frobenius norms:

$$\|\overline{\overline{{\bf B}}}_\ell\|_F^2\approx\|\widetilde{{\bf B}}_\ell\|_F^2.$$

The conjecture is motivated by the fact that the associated vector sequences both extend the Krylov space beyond $\mathcal{R}({\bf V}_m)$ and are orthogonal to the approximate dominant singular vectors. It is used to estimate the contribution of ${\bf T}\tau{\bf B}_{m+\ell}{\bf Z}_c$ and is tested numerically in the paper; no proof or resolution is provided.