The higher-dimensional bootstrapping conjecture for lattice walks

Let ${\mathfrak{S}}$ be a set of small steps in $\mathbb{Z}^d$ for some dimension $d$, and let $R$ be an open region in $\mathbb{Z}^d$ bounded by rational hyperplanes. Define

$${\mathfrak{T}}={\mathfrak{S}}\cup\{s_0\},\qquad {\mathfrak{U}}={\mathfrak{S}}\cup\{s_1,s_2\}.$$

Here $s_0$ is a step away from any boundary of $R$, and $s_1,s_2$ is a pair of steps without drift towards any boundary. Let $\beta_{\mathfrak{S}}$, $\beta_{\mathfrak{T}}$, and $\beta_{\mathfrak{U}}$ denote the exponential growth of walks in $R$ on the respective step sets. The higher-dimensional bootstrapping conjecture. The lower bounds

$$\beta_{\mathfrak{S}}+1\leq\beta_{\mathfrak{T}},\qquad \beta_{\mathfrak{S}}+2\leq\beta_{\mathfrak{U}}$$

should hold, as suggested by Lemmas onestep and twostep. This proposes that the planar bootstrapping arguments extend to lattice walks in regions of arbitrary dimension bounded by rational hyperplanes.