Lemke Oliver–Soundararajan conjecture for consecutive prime residue patterns

Let $q\ge3$ be a positive integer, let $a$ and $b$ be reduced residue classes modulo $q$, and set $h\equiv b-a\pmod q$. Define

$$\alpha(y)=1-\frac{q}{\varphi(q)\log y},\qquad \epsilon_q(a,b)=\#\{0<t<h:(t+a,q)=1\}-\frac{\varphi(q)}{q}h.$$

Let $\mathfrak{S}_{q,0}$ denote the inclusion–exclusion modification of the modified singular series introduced in the preceding context, and define $\mathcal{D}(a,b;y)$ by the sum in the statement below. Lemke Oliver–Soundararajan conjecture. The consecutive-prime counting function satisfies

$$\pi(x;q,(a,b))\sim\frac1q\int_2^x\alpha(y)^{\epsilon_q(a,b)}\left(\frac{q}{\varphi(q)\alpha(y)\log y}\right)^2\mathcal{D}(a,b;y)\,dy,$$

where

$$\mathcal{D}(a,b;y)=\sum_{\substack{h>0\h\equiv b-a\pmod q}}\sum_{\mathcal{A}\subset\{0,h\}}\sum_{\substack{\mathcal{T}\subset[1,h-1]\\(t+a,q)=1\ \forall t\in\mathcal{T}}}(-1)^{|\mathcal{T}|}\mathfrak{S}_{q,0}(\mathcal{A}\cup\mathcal{T})\left(\frac{q}{\varphi(q)\alpha(y)\log y}\right)^{|\mathcal{T}|}\alpha(y)^{h\varphi(q)/q}.$$

This conjecture is intended to explain the observed biases among consecutive prime residue patterns modulo $q$, while incorporating lower-order terms omitted in the original heuristic. The source attributes it to Lemke Oliver and Soundararajan and does not provide evidence of resolution.