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  $(x_{n}-x\, y$) $\rightarrow$ $0$
whenever $y$ in $E$.

(That is to say, $(E,(.,.))$ is *weakly sequentially complete*.) 

and

(**DPG**) If $(x_{n}\, y)$ $\rightarrow$ $0$ for every $y$ in $E$,
then $(x_{n}\, x_{n})$ $\rightarrow$ $0$.

(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 $(E,(.,.))$ .)