Definition of $k$-precosymplectic manifold

Question

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such that $\mathrm{rk}\omega = 2r$ and $\omega^r\wedge\eta \neq 0$.

On the other hand, a $k$-cosymplectic manifold is a tuple $(M,\omega^1,\dotsc,\omega^k,\eta^1,\dotsc,\eta^k,\mathcal{V})$ where $M$ is a smooth manifod of dimension $k+m(k+1)$, $\omega^\alpha$ is a closed 2-form on $M$, $\eta^\alpha$ is a closed 1-form on $M$ for every $\alpha = 1,\dotsc,k$ and $\mathcal{V}$ is an integrable distribution of dimension $mk$ such that

1. $\eta^1\wedge\dotsb\wedge\eta^k \neq 0$.
2. $\eta^\alpha\vert_\mathcal{V} = 0$, $\omega^\alpha\vert_{\mathcal{V}\times\mathcal{V}}\neq 0\qquad\forall\alpha = 1,\dotsc,k$.
3. $\left(\bigcap_{\alpha = 1}^k\ker\omega^\alpha\right)\cap\left(\bigcap_{\alpha = 1}^k\ker\eta^\alpha\right) = \{0\}.$
4. $\dim\left(\bigcap_{\alpha = 1}^k\ker\omega^\alpha\right) = k.$

I would like to know how should I modify the definition of $k$-cosymplectic manifold to obtain a definition of $k$-precosymplectic manifold of rank $2r$ such that when I put $k=1$ I get the definition of a precosymplectic manifold of rank $2r$.

Thank you.