Polynomial Somos sequences II

It was proved in [1] that for $k=4,5,6,7$ the elements of the Somos-$k$ sequence defined by the recurrence
$$S_k(n+k)S_k(n)=\sum_{1\leqslant i\leqslant k/2}\alpha_i x_0\dots x_{k-1}S_k(n+k-i)S_k(n+i)$$
and initial values $S_k(j)=x_j$ ($j=0,\dots,k-1$) are polynomials in the variables $x_0,\dots,x_{k-1}$. The unit powers of the variables $x_j$ in the factors \linebreak $\alpha_i x_0\dots x_{k-1}$ can be reduced. In this paper, we find the smallest values of these powers, at which the polynomiality of the above sequence is preserved.