  This paper is about non-holomorphic isometric immersions of Kaehler manifolds
into Euclidean space $f\colon M^{2n}\to\R^{2n+p}$, $p\leq n-1$, with low
codimension $p\leq 11$. In particular, it addresses a conjecture proposed by J.
Yan and F. Zheng. The claim that if the index of complex relative nullity of
the submanifold satisfies $\nu_f^c<2n-2p$ at any point, then $f(M)$ can be
realized as a holomorphic submanifold of a non-holomorphic Kaehler submanifold
of $\R^{2n+p}$ of larger dimension and some large index of complex relative
nullity. This conjecture had previously been confirmed by Dajczer-Gromoll for
codimension $p=3$, and then by Yan-Zheng for $p=4$. For codimension $p\leq 11$,
we already showed that the pointwise structure of the second fundamental form
of the submanifold aligns with the anticipated characteristics, assuming the
validity of the conjecture. In this paper, we confirm the conjecture until
codimension $p=6$, whereas for codimensions $7\leq p\leq 9$ it is also possible
that the submanifold exhibits a complex ruled structure with rulings of a
specific dimension. Moreover, we prove that the claim of the conjecture holds
for codimensions $7\leq p\leq 11$ albeit subject to an additional assumption.
