Leibman's connected orbit-closure conjecture for polynomial sequences

Let $(X=G/\Gamma,a)$ be a totally minimal nilsystem, let $p_1,\ldots,p_d\in\mathbb{Z}[x]$ satisfy $p_i(0)=0$, and define the polynomial sequence $g(n)=a^{p_1(n)}\otimes\cdots\otimes a^{p_d(n)}$ in $G^d$. Let $\Delta_{X^d}=\{(x,\ldots,x):x\in X\}$ be the diagonal.

Leibman's conjecture. The orbit closure

$$\overline{\{g(n)\Delta_{X^d}:n\in\mathbb{Z}\}}$$

is connected.

The paper presents this as a special case of Leibman's Conjecture 11.4 and proves that it is equivalent to the polynomial odd-recurrence conjecture above. The conjecture remains open in the supplied source.