The minimizer-set conjecture for the PNS-free segment

Let $S$ be the effective domain of fourth order four dimensional Hankel tensors under the symmetric assumptions $v_j=v_{12-j}$ for $j=0,\ldots,5$ and $v_4=v_8=1$, and let $M_0$ and $N_0$ be the functions introduced on $S$ such that a tensor is SOS if and only if $v_0\geq M_0$ and PSD if and only if $v_0\geq N_0$. Define

$$L=\{(v_2,v_6,v_1,v_3,v_5)^\top=(1,1,t,t,t)^\top:t\in[-1,1]\}.$$

The minimizer-set conjecture. The segment $L$ is the minimizer set of both $M_0$ and $N_0$.

The segment has already been proved to be PNS-free, meaning that the SOS and PSD thresholds coincide there. The conjecture, based on numerical experiments, identifies all minimizers of both threshold functions; the source does not provide a proof or resolution.