McCranie's prime-quadruplet characterization of jump-condition solutions

Let $\varphi$ denote Euler's totient function and $\sigma$ the sum-of-divisors function. A prime quadruplet is a tuple of primes of the form

$$(p,p+2,p+6,p+8).$$

McCranie's conjecture. Every composite integer $n$ satisfying

$$\varphi(n+12)=\varphi(n)+12$$

and

$$\sigma(n+12)=\sigma(n)+12$$

is of the form $n=p(p+8)$, where $(p,p+2,p+6,p+8)$ is a prime quadruplet with $p\geq 5$.

The conjecture would characterize all composite solutions to the simultaneous jump conditions and imply Stephan's corrected congruence claim $n\equiv65\pmod{72}$. It connects these arithmetic-function conditions with the Hardy–Littlewood prime $4$-tuple conjecture; the paper proves the characterization for squarefree semiprimes and rules out prime-power solutions, while the general case remains open.