Let $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] properties:
(WSC) If the sequence { ${(x_{n},\ y)}$ } is Cauchy for each $y$ in $E$, then there exists some $x$ in $E$ such that $\left(x_{n}-x,y\right)$ $\rightarrow$ $0$ whenever $y$ in $E$.
(That is to say, $(E,(.,.))$ is weakly sequentially complete.)
and
(DPG) If $\left(x_{n},y\right)$ $\rightarrow$ $0$ for every $y$ in $E$, then $\left(x_{n},x_{n}\right)$ $\rightarrow$ $0$.
(That would be sort of "Dunford-Pettis & Grothendieck'' property for indefinite inner product spaces.)
Suppose also that $|E|$ is "big enough'' (at least $|E|$ $>|\mathbb{R}|$).
Conjecture. $(E,(.,.))$ contains an infinite-dimensional Hilbert subspace.
(That is, there exists a linear isometry from $(\ell^{2},<.,.>)$ into $(E,(.,.))$ .)
