The universal primorial form of the twin prime conjecture

Let $p_n$ be the $n$-th prime, let

$$p_n\#=\prod_{i=1}^{n}p_i,$$

and let $K\in\left\{\frac{N}{2}:N=1,2,\ldots\right\}$. A universal primorial is an odd positive integer of the form $Kp_n\#+g$, where

$$g=\begin{cases}\pm1&\text{for even }N,\\2\text{ or }4&\text{for odd }N.\end{cases}$$

Universal primorial conjecture. There are expected to be infinitely many numbers $x=Kp_n\#+g$ such that both $x$ and $x+2$ are prime.

Since every positive integer can be written as a universal primorial, the paper states that this is equivalent to the Twin Prime Conjecture.