fa.functional analysis - Hilbert subspaces of indefinite inner product spaces

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

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

and

(DPG) If $left(x_{n},yright)$ $rightarrow$ $0$ 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 $(E,(.,.))$ .)