Non-regularity of polar spaces induced from proper dual subspaces

Let $q$ be the form described in the paper's construction, let $V$ be its underlying vector space, and let $V'$ be a proper subspace of $V^*$ such that

$$\ker(V')=\{0\}.$$

Let the induced form on $V\oplus V'$ define a polar space.

Non-regularity conjecture. The polar space associated to this induced form is non-regular.

The conjecture is motivated by the failure of the identity needed to reproduce the regularity proof for the original quadric when $V^*$ is replaced by a proper subspace $V'$. The supplied text gives no resolution status.