Hudson's conjecture on completely multiplicative functions of length three

Let $F_{2,3}$ denote the set of completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ whose maximal run of consecutive $+1$ values has length $3$. For primes $p\in\{5,7,11,13,53\}$, let $f_p^{\pm}$ be the modified Legendre symbols defined by $f_p^{\pm}(p')=\left(\frac{p'}{p}\right)$ for $p'\ne p$ and $f_p^{\pm}(p)=\pm1$. Let $g_1,g_2,g_3$ be completely multiplicative functions with $g_1(p)=g_2(p)=(-1)^{(p-1)/2}$ for $p\ne2$, $g_1(2)=-g_2(2)=+1$, $g_3(p)=+1$ for $p\ne2$, and $g_3(2)=-1$. Hudson's conjecture. The set $F_{2,3}$ consists precisely of the $13$ functions $F_{2,3}=S\cup T$, where $S=\{f_p^+:p\in\{5,7,11,13,53\}\}\cup\{f_p^-:p\in\{5,7,11,13,53\}\}$ and $T=\{g_1,g_2,g_3\}$. This is a classification conjecture for the much more difficult length-three case; the supplied text gives no resolution status.