Modified Polignac's conjecture for number-trail prime gaps

Let $\mathcal{D}_k^1=L_{\infty}(p_{k+1})-L_{\infty}(p_k)$ be the prime gaps along the number trail, and let $\mathbb{N}$ denote the natural numbers. Modified Polignac's conjecture. For every $N\in\mathbb{N}\setminus\{1,3,5\}$, there are infinitely many $k$ such that $\mathcal{D}_k^1=N$. The preceding result shows that $\mathcal{D}^1$ takes arbitrarily large values but never takes $3$ or $5$; the conjecture proposes infinitude for every remaining natural value except $1$, whose treatment is also excluded by the stated formulation. Its resolution is not given in the source.