Elimination of extremal index zeroes from generic paths of closed 1-forms

Let $\alpha$ be a Morse closed $1$-form of a smooth $n$-dimensional manifold $M$. The zeroes of $\alpha$ of index $0$ or $n$ are called \emph{centers}. It is known that every non-vanishing de Rham cohomology class $u$ contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path $(\alpha_t)_{t\in [0,1]}$ of closed $1$-forms in a fixed class $u\neq 0$ such that $\alpha_0, \alpha_1$ have no centers, can be modified relatively to its extremities to another such path $(\beta_t)_{t\in [0,1]}$ having no center at all.