Kaczorowski's positive-density conjecture for prime-number races

Let $k$ be a modulus, let $\varphi(k)$ denote Euler's totient, and let $a_1,\ldots,a_{\varphi(k)}$ be a permutation of the reduced residue classes modulo $k$. Write $\pi(m,k,a)$ for the number of primes up to $m$ congruent to $a$ modulo $k$. Kaczorowski's conjecture. For each permutation $a_1,\ldots,a_{\varphi(k)}$ of the reduced residue classes modulo $k$, the set of integers $m$ such that

$$\pi(m,k,a_1)<\pi(m,k,a_2)<\dots<\pi(m,k,a_{\varphi(k)})$$

has positive lower density. This predicts that every ordering of the prime-counting functions occurs with positive frequency. The source attributes the proposal to Kaczorowski and gives no resolution status.