Parity conjecture for the Möbius function on shifted primes

Let $S=(q_1,\dots,q_t)$ be a finite sequence of primes and define $\nu_S(n)=(\nu_{q_1}(n),\dots,\nu_{q_t}(n))$. Say that $n$ is $S$-squarefree if $q^2\nmid n$ for every prime $q\notin S$, and define

$$\mu_S(n)=\begin{cases}(-1)^{\sum_{q\mid n,\ q\notin S}1}&\text{if $n$ is $S$-squarefree},\\0&\text{otherwise.}\end{cases}$$

The shifted-prime parity conjecture. For non-zero $t$, $S=\{q_1,\dots,q_t\}$, and natural numbers $e_1,\dots,e_t$,

$$\sum_{p\le x\atop \nu_S(p-1)=(e_1,\dots,e_t)}\mu_S(p-1)=o(\pi(x)),$$

where the $o$-constant depends at most on $S,e_1,\dots,e_t$ and $H$. This generalizes the belief that $p-1$ has no preference for an even or odd number of prime factors; the source describes it as deep and gives no resolution.