Toroczkai's independent-edge conjecture for prime gap graphs

Let $p_n$ denote the $n$-th prime number, with $p_0=1$. A prime gap graph on $n$ vertices is a simple graph whose vertex degrees are $p_1-p_0,\dotsc,p_n-p_{n-1}$; two edges are independent when they have no common vertex. Toroczkai's independent-edge conjecture. In every prime gap graph on $n$ vertices, there exist

$$\frac{p_{n+1}-p_n}{2}$$

independent edges. The supplied context does not state whether this assertion has been proved or remains open.