Dickson's conjecture for residue class patterns of consecutive primes

For $q,m \in \mathbb{N}$, let $p_n$ denote the $n$-th prime and define

$$\pi(x;q,\mathbf{a})=\#\{p_n \leq x:p_{n+i-1} \equiv a_i \pmod{q} \text{ for all }i=1,2,\ldots,m\},$$

for $\mathbf{a} \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$. Dickson's conjecture. For any $q,m \in \mathbb{N}$ and $\mathbf{a} \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$,

$$\pi(x;q,\mathbf{a}) \to \infty \quad\text{as } x \to \infty.$$

Equivalently,

$$\#\left\{\mathbf{a} \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times: \pi(x;q,\mathbf{a}) \to \infty \text{ as } x \to \infty\right\}=\varphi(q)^m.$$

This is presented as a consequence of Dickson's conjecture and asserts that every admissible residue pattern occurs infinitely often among consecutive primes. The statement is not proved in the paper and is therefore open.