Moment bounds for vector martingale quadratic forms without the convergence assumption

Let $(\varepsilon_n)$ be a martingale difference sequence satisfying the homogeneity condition

$$\mathbb{E}[\varepsilon_{n+1}^2\mid \mathcal{F}_n]=\sigma^2 \quad \text{a.s.}$$

and assumption $(H_p)$ introduced in the preceding theorem, for some integer $p\geq 1$. Define $f_k$, $M_k$, $S_{k-1}$, $S_k$, and $d_n$ as in that theorem. Moment-bound conjecture. Almost surely,

$$\sum_{k=1}^{n}f_k\bigl(M_k^{t}S_{k-1}^{-1}M_k\bigr)^p=\mathcal{O}(\log d_n)$$

and

$$\sum_{k=1}^{n}\left[\bigl(M_k^{t}S_{k-1}^{-1}M_k\bigr)^p-\bigl(M_k^{t}S_k^{-1}M_k\bigr)^p\right]=\mathcal{O}(\log d_n).$$

The conjecture would extend the paper's convergence estimates to the full vector problem without assuming the technical condition that all eigenvalues of $S_n$ grow at the same rate. The statement is presented as unresolved in the source.