The oblivious adversary conjecture for Oja's algorithm

Let $T$ be sufficiently large and let $\boldsymbol{A}_1,\dots,\boldsymbol{A}_T$ be a sequence of matrices in the space of symmetric $n\times n$ matrices, each with spectral norm at most $1$. Fix a step size $\mu$.

Oja's linear-regret conjecture. There exists such a sequence $\boldsymbol{A}_1,\dots,\boldsymbol{A}_T$, chosen obliviously and independently of Oja's initial random vector, for which Oja's algorithm incurs regret linear in $T$ with at least constant probability.

The question concerns whether Oja's algorithm can guarantee sublinear regret in the oblivious setting while storing only a vector and receiving only first-order information. The statement asserts that, for every fixed step size and sufficiently large horizon, an obliviously chosen bounded sequence can nevertheless force linear regret with nonvanishing probability.