  Embedded in an ohmic environment, the Josephson current peak can transfer
part of its weight to finite voltage and the junction becomes resistive. The
dissipative environment can even suppress the superconducting effect of the
junction via a quantum phase transition occuring when the ohmic resistance
$R_s$ exceeds the quantum resistance $R_{q}=h/(2e)^2$. For a topological
junction hosting Majorana bound states with a $4 \pi$ periodicity of the
superconducting phase, the phase transition is shifted to $4 R_{q}$. We
consider a Josephson junction mixing the $2 \pi$ and $4 \pi$ periodicities
shunted by a resistor, with a resistance between $R_q$ and $4 R_q$. Starting
with a quantum circuit model, we derive the non-monotonic temperature
dependence of its differential resistance resulting from the competition
between the two periodicities; the $4 \pi$ periodicity dominating at the lowest
temperatures. The non-monotonic behaviour is first revealed by straightforward
perturbation theory and then substantiated by a fermionization to exactly
solvable models when $R_s=2R_{q}$: the model is mapped onto a helical wire
coupled to a topological superconductor when the Josephson energy is small and
to the Emery-Kivelson line of the two-channel Kondo model in the opposite case.
