Lins triality formulas for prisms and antiprisms

Let ${\cal M}$ be a map, and denote the Euler characteristics of ${\cal M}$, $phial({\cal M})$, and $skew({\cal M})$ by $\chi({\cal M})$, $\chi_p({\cal M})$, and $\chi_s({\cal M})$, respectively. Lins-triality conjecture. For $Prism_m$, one has $\chi_s(Prism_m)=\gcd(m,4)-m$, with $skew(Prism_m)$ oriented exactly when $m$ is even, and

$$\chi_p(Prism_m)=2+\gcd(m,4)-2m=\chi(Prism_m)+\chi_s(Prism_m)-m,$$

with $phial({\cal M})$ non-oriented. For $APrism_m$, one has $\chi_s(APrism_m)=1+\gcd(m,3)-2m$ with $skew({\cal M})$ non-oriented, and

$$\chi_p(APrism_m)=3+\gcd(m,3)-2m=\chi(APrism_m)+\chi_s(APrism_m),$$

with $skew(APrism_m)$ oriented. The formulas were checked up to $n=100$, but the supplied text does not establish them in general.