Toroczkai's existence conjecture for prime gap graphs

Let $p_n$ be the $n$-th prime, with $p_0=1$, and let

$$\mathrm{PD}_n=\left(p_{\ell}-p_{\ell-1}\right)_{\ell=1}^{n}$$

be the first $n$ prime gap sequence. A prime gap graph on $n\geq 2$ vertices is a simple graph whose vertex degrees are exactly the entries of $\mathrm{PD}_n$. Toroczkai's existence conjecture. For every $n\geq 2$, there exists a prime gap graph on $n$ vertices.

The conjecture asserts graphicality of every finite initial segment of the prime gap sequence; the paper establishes explicit unconditional graphicality only beyond a very large threshold, so the full assertion remains open.