The Skewes-number conjecture for admissible prime tuples

Let $P=(p,p+a_1,p+a_2,\ldots,p+a_k)$ be an admissible prime $k$-tuple, with associated counting function $\pi_P(n)$ and Hardy–Littlewood constant $C_{a_1,a_2,\ldots,a_k}$. A Skewes number for $P$ is the first prime $p\in P$, if such a prime exists, at which

$$\pi_P(n)\geq C_{a_1,a_2,\ldots,a_k}\int_2^n\frac{dt}{\log^{k+1}t}.$$

Skewes-number conjecture. Every admissible prime $k$-tuple has a Skewes number. The conjecture is based on the computational results in the paper; no proof or disproof is supplied.