N-bonacci growth conjecture for iterated Wronskian brackets

Let $d\geqslant 1$ be the dimension of the base $\mathbb{R}^d$, let $k\geqslant 1$ be a differential order, and set

$$N=\binom{d+k}{d}.$$

Let $W_d^k$ denote the Wronskian $N$-ary bracket on $\mathbb{R}[x^1,\ldots,x^d]$. N-bonacci growth conjecture. There exists an $N$-tuple of monomials $(m_1(\boldsymbol{x}),\ldots,m_N(\boldsymbol{x}))\in\mathbb{R}[x^1,\ldots,x^d]$ such that the iteration

$$m_n(\boldsymbol{x}):= W_d^k(m_{n-N},\ldots,m_{n-1})$$

yields nonzero monomials for every $n\in\mathbb{N}$, with their total degrees, or degrees in some or every variable, growing asymptotically as fast as the $N$-bonacci numbers. The conjecture concerns whether the upper growth rate $\alpha_N^n$, where $\alpha_N$ is the exponential growth constant of the $N$-bonacci sequence, can be attained.