The mod 4 characterization of continued fractions with $\mu(r)=-1$

Let $p$ be the parameter defining the family $H(p)$, and write a continued fraction as $r=[pk_1,2m_1,pk_2,2m_2,\dots,pk_{\ell+1}]$. Two such continued fractions are (mod 4)-equivalent if one can be transformed into the other by the five stated operations and their inverses. Let $\mu(r)$ be the associated invariant.

Mod 4 characterization conjecture. $\mu(r)=-1$ if and only if $r$ is (mod 4)-equivalent to either $[p]$ or $[3p]$.

This conjecture gives a classification of the continued fractions for which the invariant takes its smallest stated value. The source presents it as plausible on the basis of computations; no resolution is supplied.