The inhomogeneous-shift classification conjecture for multiplicative functions

Let $f:\mathbb{N}\to\mathbb{Z}$ be multiplicative and satisfy

$$\sum_{p:\,|f(p)|\neq 1}\frac{1}{p}=\infty.$$

For nonzero $a,b\in\mathbb{Z}$, define

$$\mathcal{N}_{f,a,b}:=\{n\in\mathbb{N}:f(n+a)=f(n)+b\},$$

and let $\delta$ denote logarithmic density. Also define

$$\mathcal{E}_f:=\{p:|f(p)|\neq p\}.$$

Inhomogeneous-shift classification conjecture. If $\delta(\mathcal{N}_{f,a,b})\neq 0$ for some nonzero $a,b\in\mathbb{Z}$, then

$$\sum_{p\in\mathcal{E}_f}\frac{1}{p}<\infty.$$

Moreover, if $f$ is completely multiplicative, then there is a divisor $d$ of $a$ such that

$$f(d)\mid b \qquad\text{and}\qquad (a/d)\mid (b/f(d)).$$

This conjecture aims to classify multiplicative functions for which an inhomogeneous shift equation has positive logarithmic density. The supplied text does not state that it has been proved or disproved.