Pivot-orbit containment conjecture for the family ${\cal F}^{n,t}$

Let $n$ and $t$ satisfy $0\leq t\leq n-1$, and let ${\cal F}^{n,t}$ be the family of Boolean functions defined by

$$f^{n,t}=\sum_{i=0}^{t-1}\sum_{j=t}^{n-1}x_ix_j+\sum_{i=t}^{n-2}\sum_{j=i+1}^{n-1}x_ix_j+a(x_0,x_1,\ldots,x_{n-1}),$$

where $\deg(a)\leq 1$, together with arbitrary Boolean functions of $x_0,\ldots,x_{t-1}$; the pivot orbit of a Boolean function is the set of functions obtained from it by successive pivot operations. Pivot-orbit containment conjecture. If $f\in{\cal F}^{n,t}$, then its pivot orbit is contained in

$$\bigcup_{k=0}^{n-1}{\cal F}^{n,k}.$$

This claim concerns the invariance of the constructed family under pivot operations, up to changing the parameter $t$. The supplied excerpt does not state whether the claim has been proved or remains open.